Loren Coquille

Sylvain Courte

François Dahmani

Pierre Dehornoy

Philippe Eyssidieux

Jean Fasel

Stéphane Guillermou

Vincent Lafforgue

Dietrich Häfner

Jean-Baptiste Meilhan

Hervé Pajot

graduate students, including Clément Berat

Gabriella Clemente

Clément Debin

Bruno Laurent

David Leturcq

Louis-Clément Lefèvre

Alejandro Rivera

Deng Ya

Zhizhong Huang

Jian Xiao

Peng Xiaojun Wu

Nicola Cavallucci

Simone Calamai

Jiaming Chen

Jiao He

Si Chen

Yahui Leng

Yu Xie

Chenlei Yu

Honghao Gao

Bingyu Zhang

Chen- Min Sun

Chao Qian

Yuexun Wang

Jun Wang

Jie Liu

Jie Shen

Sheng Rao

Keywords: 2010 Mathematics Subject Classification. Primary 53C20, 53C21; Secondary 49Q05, 58E12 Thurston's Geometrisation Conjecture, Positive Scalar Curvature, Contractible 3-manifolds, Simply-connectedness at infinity, Fundamental group at infinity, Loop Lemma, Handlebody, Genus One 3-manifolds, Minimal Surface Technique, Minimal Surface Equation, Plateau's Problem, Stable Minimal Surface, Lamination, Extrinsic Cohn-Vesson's Inequality, Aspherical manifolds, Gromov-Lawson Conjecture

The purposes of this thesis is to understand spaces which carry metrics of positive scalar curvature. There are several topological obstructions for a smooth manifold to have a complete metric of positive scalar curvature. Our goal is to find all obstructions for contractible 3-manifolds and closed 4-manifolds.

Demailly. I enjoyed the lunch time with him. I am thankful to my collaborator, Long Li,

In dimension 3, we are concerned with the question whether a complete contractible 3-manifold of positive scalar curvature is homeomorphic to R 3 . The topological structure of contractible 3-manifolds could be complicated. For example, the Whitehead manifold is a contractible 3-manifold which is not homeomorphic to R 3 . We first prove that the Whitehead manifold does not carry a complete metric of positive scalar curvature. This result can be generalised to the so-called genus one case. Precisely, we show that no contractible genus one 3-manifold admits a complete metric of positive scalar curvature.

We then study the fundamental group at infinity, π 8 1 , and its relationship with the existence of positive scalar curvature metric. The fundamental group at infinity of a manifold is the inverse limit of the fundamental groups of complements of compact subsets. In this thesis, we give a partial answer to the above question. We prove that a complete contractible 3-manifold with positive scalar curvature and trivial π 8

1 is homeomorphic to R 3 .

Finally, we study closed aspherical 4-manifolds. Together with minimal surface theory and the geometrisation conjecture, we show that no closed aspherical 4-manifold with nontrivial first Betti number carries a metric of positive scalar curvature.

Résumé

Un des objectifs de ce mémoire est de comprendre les espaces munis de métriques complète de courbure scalaire positive. Il y a plusieurs obstructions topologiques à l'existence d'une métrique complète de courbure scalaire positive. Notre but est de trouver toutes les obstructions pour les variétés contractiles de dimension 3 et les variétés fermées de dimension 4.

En dimension 3, nous considérons la question de savoir si une variété contractile complète de courbure scalaire positive est homéomorphe à R 3 . La structure topologique des variétés contractiles de dimension 3 est assez compliquée. Par exemple, Whitehead a construit une variété dimension 3 contractile qui n'est pas homéomorphe à R 3 . Nous prouvons, tout d'abord, que la variété de Whitehead n'a pas de métrique complète de courbure scalaire positive. Ce résultat peut être généralisé au cas dit de genre un. Précisément, nous montrons qu'aucune variété contractile de dimension 3 et de genre un ne possède de métrique complète de courbure scalaire positive.

Nous étudions ensuite le groupe fondamental à l'infini, π 8 1 , et son lien avec l'existence d'une métrique de courbure scalaire positive. Le groupe fondamental à l'infini d'une variété est la limite projective des groupes fondamentaux des complémentaires des sousensembles compacts. Dans ce mémoire, nous apportons une réponse partielle à la question évoquée plus haut. Nous prouvons qu'une variété complète de dimension 3 de courbure scalaire positive dont le groupe π 8 1 est trivial est homéomorphe à R 3 . Enfin, nous étudions les variétés fermées asphériques de dimension 4. En utilisant la théorie des surfaces minimales et la conjecture de géométrisation, nous montrons qu'aucune variété fermée asphérique de dimension 4 avec un premier nombre de Betti non trivial ne possède de métrique à courbure scalaire positive.

vi Acknowledgement I would like to thank my advisor Gérard Besson for all of his guidance throughout the time in Grenoble. He suggested this great project and generously shared many amazing ideas with me. Over the years, he has led my way from a student to a junior researcher though patient and dedicated suggestions. Without his constant encouragement, this dissertation may not be completed. Beside Mathematics, I respect him for his kindness and humbleness. I also thank Laurent Bessières, Christoph Böhm, Gilles Carron, Xiuxiong Chen, Jian Ge, Misha Gromov, Bernhard Hanke, Laurent Hauswirth, Ilaria Mondello, Yuguang Shi, Samuel Tapie, Richard Thom, Wilderich Tuschmann, Marina Ville, Burkhard Wilking for their interests in my thesis and for their enlightening conversations. I also thank my former advisors Yuxiang Li and Pin Yu in Tsinghua University for the encouragement and inspiration to pursue research. Beside, I am grateful to Jean-Pierre vii I also thank Laurent Mazet, Blaine Lawson for spending time and effort on reveiwing my thesis. They made many helpful and insightful suggestions that improved the exposition of my thesis. I am thankful to Yunhui Wu for pointing out an improvement of my results. In addition, I thank Louis Funar, Sylvain Maillot, David Gabai and Harold Rosenberg for their enlightening discussions and sharing their ideas .

Introduction

Riemannian geometry aims to study Riemannian manifolds which are smooth manifolds with metric structures. One of the fundamental questions is to understand the relationship between the curvature, which is locally defined, and the global properties of smooth manifolds. The earliest result is the classical theorem of Gauss and Bonnet, which links the curvature and the Euler number, a topological invariant. This theorem implies, for example, that any compact surface of genus g ą 0 has no metric of positive curvature.

In higher dimensions, the existence of metrics of positive curvature becomes much more complicated, because there are several topological obstructions for a smooth manifold to have a complete metric of positive curvature.

We then take 3-manifolds to explain this fact.

pM 3 , gq

The compact case The non-compact case

K ą 0 S 3 {Γ R 3 Ric ą 0 R 3
Scal ą 0 p# k i"1 S 3 {Γ i q#p# l j"1 S 1 ˆS2 q ?

Table 1.

For a compact Riemannian 3-manifold, there is a unique short-time solution to the so-called (normalized) Ricci flow, introduced by Hamilton [START_REF] Hamilton | Three-manifolds with positive ricci curvature[END_REF]. If the manifold has positive Ricci curvature, the short-time solution can be extended to be the long-time solution. The limit of this flow is a metric of constant sectional curvature. That is to say, if a compact 3-manifold has positive Ricci curvature, then it is homeomorphic to the quotient S 3 {Γ of the sphere S 3 by a finite subgroup Γ Ă Op4q. Such a quotient is called a spherical 3-manifold.

The next major contribution to the subject was made by Perelman [START_REF] Perelman | The entropy formula for ricci flow and its geometric applications[END_REF][START_REF] Perelman | Ricci flow with surgery on three-manifolds[END_REF][START_REF] Perelman | Finite extinction time for the solutions to the ricci flow on certain threemanifolds[END_REF] who developed the Ricci flow with surgery. One surprising and beautiful result of this study is a proof that a compact 3-manifold of positive scalar curvature is homeomorphic to a connected sum of some spherical 3-manifolds and some copies of S 1 ˆS2 (See also [BBB `10] and [START_REF] Tian | Ricci flow and the Poincaré conjecture[END_REF]). Its generalization to the non-compact case is due to Bessières, Besson, and Maillot [START_REF] Bessières | Ricci flow on open 3-manifolds and positive scalar curvature[END_REF].

For the non-compact case, the first result shown by Gromoll and Meyer [START_REF] Gromoll | On complete open manifolds of positive curvatur[END_REF] was that a complete non-compact 3-manifold of positive sectional curvature is homeomorphic to R 3 . The next step in the development of this subject is due to Schoen and Yau [START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF] who used minimal surfaces theory and the splitting theorem [START_REF] Cheeger | The splitting theorem for manifolds of nonnegative ricci curvature[END_REF] to study the topology of 3-manifolds. They proved that a complete non-compact 3-manifold of positive Ricci curvature is homeomorphic to R 3 . Although all of these works are very impressive, they still left the open question (See Problem 27 in [START_REF] Shing | Problem section[END_REF]):

How to classify non-compact 3-manifolds with positive scalar curvature, up to diffeomorphism?

The goal now is to find all obstructions and to characterize all open 3-manifolds with positive scalar curvature. Although Gromov-Lawson [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF] and Schoen-Yau [START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF] gave several topological obstructions, all those obstruction classes both vanish for contractible 3-manifolds.

Let us consider contractible 3-manifolds. For example, R 3 admits a complete metric g 1 of positive scalar curvature, where

g 1 " 3 ÿ i"1 pdx i q 2 `p 3 ÿ i"1 x i dx i q 2 .
So far, it is the only known contractible 3-manifold which admits a complete metric of positive scalar curvature. This suggests the following question: Is any complete contractible 3-manifold of positive scalar curvature homeomorphic to R 3 ?

A complete contractible 3-manifold of uniformly positive scalar curvature (i.e. its scalar curvature is bounded away from zero) is homeomorphic to R 3 . It was first proved by Gromov and Lawson [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF]. Recently, it was generalized by Chang, Weinberger and Yu [START_REF] Chang | Taming 3-manifolds using scalar curvature[END_REF], to contractible 3-manifolds whose scalar curvature is uniform positive outside a compact set. Using minimal surfaces theory, we further generalize it.

Theorem A. (See Theorem 3.3.12 and Theorem 1.1 in [START_REF] Wang | Simply connected open 3-manifolds with slow decay of positive scalar curvature[END_REF]) Assume that pM 3 , gq is a contractible complete 3-manifold. If there exists a number α P p´8, 2q such that lim inf rpxqÑ8 r α pxqκpxq ą 0, where κpxq is the scalar curvature of pM, gq and rpxq is the distance function from some point 0 P M to x, then M 3 is diffeomorphic to R 3 .

The proof follows the argument of Gromov and Lawson (See Corollary 10.9 in [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF]).

Contractible 3-manifolds

Thurston's Geometrisation conjecture [START_REF] Perelman | The entropy formula for ricci flow and its geometric applications[END_REF][START_REF] Perelman | Ricci flow with surgery on three-manifolds[END_REF][START_REF] Perelman | Finite extinction time for the solutions to the ricci flow on certain threemanifolds[END_REF] (See also [BBB `10] and [START_REF] Tian | Ricci flow and the Poincaré conjecture[END_REF]) shows that for a compact 3-manifold, its topology is fully determined by its homotopy type. However, the topological structure of contractible 3-manifolds is much more complicated. For example, the Whitehead manifold (constructed in [START_REF] Whitehead | A certain open manifold whose group is unity[END_REF]) is a contractible 3-manifold but not homeomorphic to R 3 .

In order to explain the construction of the Whitehead manifold, let us introduce the concept of a meridian curve. A meridian γ Ă BN of a closed solid torus N is an embedded closed curve which is null-homotopic in N but not contractible in BN . A meridian disc pD, BDq Ă pN, BN q of the solid torus N is an embedded disc whose boundary is a meridian of N . (See Definition 1.2.1)

The Whitehead manifold is constructed from the Whitehead link. Recall that the Whitehead link is a link with two components illustrated in the following figure: Choose a closed unknotted solid torus T 1 in S 3 . Its complement inside S 3 is another solid torus. Take a second solid torus T 2 inside T 1 so that the core K 2 of T 2 forms a Whitehead link with any meridian of T 1 as in the following figure. The solid torus T 2 is unknotted in S 3 . Then, embed T 3 inside T 2 in the same way as T 2 lies into T 1 and so on infinitely many times. Define the set T 8 " Ş 8 k"1 T k , called the Whitehead continuum.

The Whitehead manifold is defined as Wh :" S 3 zT 8 which is a non-compact 3manifold without boundary.

Remark. Since each T k is unknotted in S 3 , its complement N k is a solid torus. Therefore, the Whitehead manifold is an increasing union of solid tori tN k u k . Each N k is embedded inside N k`1 in the same way as T 2 lies in T 1 . This follows from the symmetry of the Whitehead link.

Variation on the construction, like changing the knot at each step k, gives a family of so-called genus one 3-manifold, introduced in [START_REF] Dr Mcmillan | Some contractible open 3-manifolds[END_REF]. Their construction is involved with the geometric index.

If N 1 Ă N are solid tori, the geometric index, IpN 1 , N q, of N 1 in N is equal to the minimal number of points of the intersection of the core of N 1 with a meridian disc of N . A genus one 3-manifold is the ascending union of solid tori tN k u, so that for each k, N k Ă IntN k`1 and the geometric index of N k in N k`1 is not equal to zero. (See Definition 1.3.8, Definition 1.3.12 and [START_REF] Garity | Contractible 3-manifolds and the double 3-space property[END_REF]) For example, Wh is a contractible genus one 3-manifold. The geometric index IpN k , N k`1 q equals two for each k, where N k is illustrated as above. Remark that R 3 is not genus one but genus zero, since it is an increasing union of 0-handlebodies (i.e. 3-balls).

An interesting question is whether the Whitehead manifold admits a complete metric of positive scalar curvature. In this thesis, we answer negatively:

Theorem B 1 . (See Theorem 1.1 of [START_REF] Wang | Contractible 3-manifolds and positive scalar curvatures (I)[END_REF]) The Whitehead manifold has no complete metric of positive scalar curvature. This result can be generalized to the genus one case.

Theorem B 2 . (See Theorem 1.2 of [START_REF] Wang | Contractible 3-manifolds and positive scalar curvatures (I)[END_REF]) No contractible genus one 3-manifold has a complete metric of positive scalar curvature.

Combining with Kazdan's work [START_REF] Kazdan | Deformation to positive scalar curvature on complete manifolds[END_REF], we generalize these results to the nonnegative scalar curvature.

Corollary. (See Corollary 6.4.3) No contractible genus one 3-manifold admits a complete metric of non-negative scalar curvature.

The existence of complete metrics of positive scalar curvature is related with the fundamental group at infinity. The fundamental group π 8 1 at infinity of a connected space is the inverse limit of the fundamental groups of complements of compact subsets (See Definition 1.1.9).

The triviality of the fundamental group at infinity is not equivalent to the simplyconnectedness at infinity(See Definition 1.1.7). For example, the Whitehead manifold is not simply-connected at infinity but its fundamental group at infinity is trivial.

We prove the following:

Theorem C. (See Theorem 1.1 of [START_REF] Wang | Contractible 3-manifolds and positive scalar curvatures[END_REF]) A contractible 3-manifold with positive scalar curvature and trivial π 8

1 is homeomorphic to R 3 . This result can also be generalised to the non-negative scalar curvature.

Corollary. (See Corollary 6.4.4) A contractible 3-manifold with non-negative scalar curvature and trivial π 8

1 is homeomorphic to R 3 . However, there are uncountably many mutually non-homeomorphic contractible 3manifolds with non-trivial π 8

1 . In Chapter 1.3, we construct such a manifold and show that this manifold has no complete metric of positive scalar curvature (See Theorem 1.3.15).

The idea of the proof of Theorem B 2

It is classical that minimal surfaces theory gives topological information about 3manifolds. This fact appeared in the articles of Schoen and Yau [START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF][START_REF] Schoen | On the structure of manifolds with positive scalar curvature[END_REF][START_REF] Schoen | Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature[END_REF] as well as Gromov and Lawson's [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF] and various other works.

For the proof of Theorem B 2 , we argue by contradiction. Suppose that pM, gq is a complete Riemannian manifold of positive scalar curvature, where M :" Ť k N k is a contractible genus one 3-manifold and the family tN k u k of solid tori is assumed as in Theorem 1.3.13. 0.2.1. Minimal surfaces and Limits. As in [START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF] and [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF], our first step is to construct minimal surfaces. Choose γ k Ă BN k as a meridian of N k (See Definition 1.2.1). Roughly, it is spanned by an embedded stable minimal disc Ω k . Its existence is ensured by the result of Meeks and Yau (See [START_REF] Meeks | Topology of three dimensional manifolds and the embedding problems in minimal surface theory[END_REF][START_REF] Meeks | The existence of embedded minimal surfaces and the problem of uniqueness[END_REF] or Theorem 3.1.8) when the boundary BN k is mean convex.

Let us consider the simplest case when Ω k converges to a connected stable minimal surface Σ.

On the one side, we show that the number of connected components of Ω k X N 1 intersecting N 0 goes to infinity as k goes to infinity (See Chapter 2.1). Therefore, there are infinitely many connected components of Σ X N 1 intersecting N 0 . By a result of Meeks and Yau (See Theorem 3.2.7 anc [START_REF] Meeks | Topology of three dimensional manifolds and the embedding problems in minimal surface theory[END_REF]), each of these components contains a definitive amount of area. Hence, Σ X N 1 has infinite area.

On the other side, since pM, gq has positive scalar curvature, not only Σ is conformally diffeomorphic to R 2 (See Corollary 3.3.11), but also its geometry is constrained by the so-called extrinsic Cohn-Vossen inequality: Theorem D. (See Theorem 3.3.10 and [START_REF] Wang | Contractible 3-manifolds and positive scalar curvatures (I)[END_REF]) Let Σ 2 Ă pM 3 , gq be a complete (non-compact) immersed stable minimal surface. If the complete manifold pM 3 , gq has non-negative scalar curvature (κpxq ě 0), then ż Σ κpxq `1{2|A| 2 dv ď 2πχpΣq where |A| 2 is the square norm of the second fundamental form of Σ. Moreover, if κ ą 0 and Σ is embedded, then Σ is a properly embedded plane.

Since the scalar curvature is bounded away zero on N 1 , this is in contradiction with the infinite area contained in Σ X N 1 .

If BN k is not mean convex, we modify the metric in a smaller tubular neighborhood of BN k so that for the new metric, it becomes mean convex. Then Ω k is stable minimal for the new metric and for the original away from the neighborhood BN k , (for example, near N k ), which is sufficient for our proof. 0.2.2. Properties of the limit surface. Generally, Ω k sub-converges to a minimal lamination L :" Ť tPΓ L t (that is, a disjoint union of some embedded minimal surfaces) instead of a single surface. It may have infinitely many components. However, each leaf L t of L is a complete (non-compact) stable minimal surface (See Theorem 4.2.3). Since pM, gq has positive scalar curvature, it is homeomorphic to R 3 (See Corollary 3.3.11).

The geometry of each leaf is influenced by the extrinsic Cohn-Vossen inequality (See Theorem D) as well as by a topological property of M , called Property P (See Definition 2.1.3). These two aspects tell us that the lamination L has the Vanishing property for tN k u k . That is to say, there is a positive integer k 0 such that for any k ě k 0 and any t P Γ, any circle in L t X BN k is null-homotopic in BN k .

The reason is as follows: Suppose that there exists a sequence tk n u of increasing integers and a sequence tL tn u of leaves so that for each n, L tn X BN kn has at least one non-nullhomotopic circle in BN kn . Similar to the property of the sequence tΩ k u k (Property P ), we know that the number of connected components of L tn X N 1 intersecting N 0 goes to infinity as k n goes to infinity (See Definition 2.1.3 and Theorem 2.1.6).

The sequence tL tn u sub-converges to some leaf L t8 in the lamination L with finite multiplicity. The sub-convergence is ensured by a result of Schoen [START_REF] Schoen | Estimates for stable minimal surfaces in three dimensional manifolds[END_REF] (See Lemma 4.1.5) and the extrinsic Cohn-Vossen inequality (See Theorem D). Therefore, L t8 X N 1 has infinitely many components intersecting N 0 . As in the above case, each component has a definite amount of area. The extrinsic Cohn-Vesson inequality gives a contradiction.

Let us explain how to deduce a contradiction from the Vanishing property. We show that for any k ě k 0 , Ω k XBN k 0 contains a closed curve which is not null-homotopic in BN k 0 (See Lemma 1.3.11). Roughly speaking, these non-nullhomotopic circles will sub-converge to some closed curve in L XBN k 0 which is not contractible in BN k 0 . It follows the fact that Ω k sub-converges to the lamination L . Therefore, some leaf of L has a non-contractible circle in BN k 0 . This is in contradiction with the above Vanishing property of L .

π 8

1 and the Vanishing property 0.3.1. Handlebodies and Property H. Let pM, gq be a complete contractible 3manifold of positive scalar curvature. It is an increasing union of closed handlebodies tN k u (See Theorem 1.1.12).

In the following, we consider that M is not homeomorphic to R 3 . We may assume that none of the N k is contained in a 3-ball (i.e. homeomorphic to a unit ball in R 3 ) in M (See Remark 1.1.8).

In the genus one case, the family tN k u has several good properties. For example, the maps π 1 pBN k q Ñ π 1 pM zN k q and π 1 pBN k q Ñ π 1 pN k zN 0 q are both injective (See Lemma 1.3.10). These properties are crucial and necessary in the study of the existence of complete metrics of positive scalar curvature. In general, the family tN k u may not have the above properties.

For example, the map π 1 pBN 0 q Ñ π 1 pM zN 0 q may not be injective. To overcome it, we use topological surgeries on N 0 and find a new handlebody to replace it. Precisely, we use the loop lemma to find an embedded disc pD, BDq Ă pM zN 0 , BN 0 q whose boundary is a non-contractible simple curve in BN 0 . The new handlebody is obtained from N 0 by attaching a closed tubular neighborhood N pDq of D in M zN 0 .

We repeatedly use topological surgeries on each N k to obtain a new family tR k u k of closed handlebodies with the following properties, called Property H (See Definition 2.2.5):

(1) the map π 1 pBR k q Ñ π 1 pR k zR 0 q is injective for k ą 0;

(2) the map π 1 pBR k q Ñ π 1 pM zR k q is injective for k ě 0;

(3) each R k is contractible in R k`1 but not contained in a 3-ball in M ; (4) there exists a sequence of increasing integers tj k u k , such that π 1 pBR k X BN j k q Ñ π 1 pBR k q is surjective.

Remark. If M is not homeomorphic to R 3 , the existence of such a family is ensured by Theorem 2.2.6. It is not unique. In addition, the union of such a family may be not equal to M . For example, if M :" Ť k N k is a contractible genus one 3-manifold, the family tN k u (assumed as in Theorem 1.3.13) satisfies the above Property H (See Lemma 2.10 in [START_REF] Wang | Contractible 3-manifolds and positive scalar curvatures (I)[END_REF] or Lemma 1.3.10). 0.3.2. The Vanishing property. In the genus one case, the geometry of a stable minimal surface is constrained by the geometric index (See Property P in [START_REF] Wang | Contractible 3-manifolds and positive scalar curvatures (I)[END_REF] or Definition 2.1.3). In the higher genus case, the behavior of a stable minimal surface is related to the fundamental group at infinity.

In order to clarify their relationship, let us introduce a geometric property, called the Vanishing property. First, we consider a complete contractible 3-manifold pM, gq of positive scalar curvature which is not homeomorphic to R 3 . As indicated above, there is an increasing family tR k u k of closed handlebodies with Property H.

A complete embedded stable minimal surface Σ Ă pM, gq is called to satisfy the Vanishing property for the family tR k u k if there is a positive integer kpΣq so that for k ě kpΣq, any circle in Σ X BR k is nullhomotopic in BR k (See Chapter 5).

If a complete stable minimal surface does not satisfy the Vanishing property for tR k u k , it gives a non-trivial element in π 8 1 pM q(See Lemma 5.2.1). As a consequence, if π 8 1 is trivial, any complete stable minimal surface in M has the Vanishing property for tR k u k (See Corollary 5.2.2). 0.3.3. The idea of the proof of Theorem C. We argue by contradiction. Suppose that a complete contractible 3-manifold pM, gq with positive scalar curvature and trivial π 8 1 pM q is not homeomorphic to R 3 . Before constructing minimal surfaces, let us introduce a notation from 3-dimensional topology.

A system of meridians of a handlebody N is a collection of g distinct meridians tγ l u g l"1

with the property that BN z š g l"1 γ l is homeomorphic to an open disc with some closed subdiscs removed (See Lemma 1.2.7). Its existence is ensured by Lemma 1.2.7.

Let tN k u k and tR k u k be as above. Since N 0 is not contained in a 3-ball (See Remark 1.1.8), the genus of N k is greater than zero. The handlebody N k has a system of meridians tγ l k u gpN k q l"1 . Roughly , there are gpN k q disjoint area-minimizing discs tΩ l k u l with BΩ l k " γ l k . Their existence is ensured by the works of Meeks and Yau [MIY80,MIY82] (See Theorem 6.28 of [START_REF] Tobias | A course in minimal surfaces[END_REF]) when the boundary BN k is mean convex.

Let us explain their existence. We construct these discs by induction on l. When l " 1, there is an embedded area-minimizing disc ] or Theorem 6.28 of [START_REF] Tobias | A course in minimal surfaces[END_REF]). Suppose that there are l disjointly embedded stable minimal discs tΩ i k u l i"1 with BΩ i k " γ i k . Our target is to construct a stable minimal surface Ω l`1 k with boundary γ l`1 k . Let us consider the Riemannian manifold pT k,l , g| T k,l q, where T k,l :"

Ω 1 k Ă N k with boundary γ 1 k (See [MIY80, MIY82
N k z š l i"1 Ω l k .
It is a handlebody of genus gpN k q ´l. For example, see the following figure.

Ω 1 k pN k , g k q γ 1 k pT k,1 , g k | T k,1 q Ω 1 k γ1 k ´Ω1 k γ1 k Figure 0.3.

The boundary of pT

k,l , g| T k,l q consists of BN k z š l i"1 γ i k and some disjoint discs tΩ i k ´ul i"1
and tΩ i k `ul i"1 . The two discs Ω i k ´and Ω i k `both come from the same minimal disc Ω i k . Therefore, the mean curvature of the boundary of pT k,l , g| T k,l q is non-negative. (See Chapter 4.2)

In addition, tγ i k u iąl is a system of meridians of the handlebody pT k,l , g| T k,l q. Then, we use the result of Meeks and Yau to find an embedded stable minimal surface Ω l`1 k Ă T k,l with boundary γ l`1 k . These discs tΩ i k u l`1 i"1 are disjoint in N k . This finishes the inductive construction.

As in the genus one case, if BN k is not mean convex, we can deform the metric in a small neighborhood of it so that it becomes mean convex.

Define the lamination L k :" š l Ω l k (i.e. a disjoint union of embedded surfaces). We show that each lamination L k intersects the compact set R 0 (Corollary 1.2.8). According to Colding-Mincozzi's theory (See Appendix B of [START_REF] Colding | The space of embedded minimal surfaces of fixed genus in a 3-manifold. iv: Locally simply connected[END_REF]), the sequence tL k u k subconverges to a lamination L :" Ť tPΛ L t in pM, gq (See Theorem 4.2.3). Note that each leaf L t is a complete (non-compact) stable minimal surface.

As indicated above, since pM, gq has positive scalar curvature and π 8 1 pM q is trivial, each leaf L t in L has the Vanishing property for tR k u k (See Lemma 5.2.1 and Corollary 5.2.2). Furthermore, the lamination L also satisfies the Vanishing property (See Corollary 5.2.4). That is to say, there exists a positive integer k 0 such that for any k ě k 0 and any t P Λ, any circle in

L t X BR k is nullhomotopic in BR k .
The reason is described as follows. We argue by contradiction. Suppose that there exists a sequence tk n u n of increasing integers and a sequence tL tn u of leaves in L satisfying that L tn X BR kn has at least one non-nullhomotopic circle(s) in BR kn for each n.

The sequence tL tn u smoothly subconverges to some leaf in L . For our convenience, we may assume that the sequence tL tn u converges to the leaf L t8 . The leaf L t8 satisfies the Vanishing property. That is to say, there is a positive integer kpL t8 q such that for k ě kpL t8 q, any circle BR k X L t8 is nullhomotopic in BR k .

However, since L tn X BR kn has some non-null-homotopic circle in BR kn , we know that for k n ą kpL t8 q, L tn XBR kpLt 8 q has a meridian of R kpLt 8 q (See Remark 2.2.7 and Corollary 1.2.6). These meridians of R kpLt 8 q will converge to a meridian of R kpLt 8 q which is contained in L t8 X BR kpLt 8 q . This is in contradiction with the last paragraph.

Let us explain how to deduce a contradiction from the Vanishing property of L . We show that if N k contains R k 0 (for k large enough), then L k X BR k 0 contains at least one meridian of R k 0 (See Corollary 1.2.8). As in the above case, these meridians of R k 0 will sub-converge to a non-contractible circle in L X BR k 0 . The Vanishing property of L gives a contradiction.

Closed Aspherical 4-manifolds

A manifold M is called aspherical if it is path-connected and all its higher homotopy groups vanish (i.e. π k pM q is trivial for k ě 2). The class of aspherical manifolds contains all hyperbolic manifolds and all manifolds with non-positive curvature.

An interesting question posed by Geroch is whether the torus T n , n ě 3, carries a metric of positive scalar curvature. This question was settled by Gromov-Lawson [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF] and [START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF][START_REF] Schoen | Positive scalar curvature and minimal hypersurface singularities[END_REF]. Generally, it is conjectured that Conjecture. No compact aspherical manifold has a metric of positive scalar curvature.

This conjecture was proved for 3-manifolds by Gromov and Lawson [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF]. In dimension four, it is confirmed for 4-manifolds which contains incompressible surfaces [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF].

In this thesis, we prove that Theorem F . No closed aspherical 4-manifolds with non-zero first Betti number has a metric of positive scalar curvature.

Note that there is a closed aspherical 4-manifold whose first Betti number vanishes (See [START_REF] John | Some examples of aspherical 4-manifolds that are homology 4-spheres[END_REF]).

We argue by contradiction. Suppose that there is a compact aspherical 4-manifold pM 4 , gq of positive scalar curvature, where the first Betti number b 1 pM 4 q is greater than zero.

Choose a circle γ Ă M 4 so that rγs has infinite order in H 1 pM 4 , Zq. We use the Poincaré duality to find a class u P H 3 pM 4 , Zq with ă u, rγs ą" 1. A theorem of Fleming-Federer (See [START_REF] Federer | Normal and integral currents[END_REF] or Chapter 7 of [START_REF] Simon | Lectures on geometric measure theory[END_REF]) tells us that there is a volumeminimizing hypersurface Σ 3 in this class. Therefore, the intersection number of γ and Σ 3 is equal to one.

Since pM, gq has positive scalar curvature, then Σ 3 admits a metric of positive scalar curvature (See Proposition 3.3.5). The manifold Σ 3 is homeomorphic to a connected sum of spherical 3-manifolds and some copies of S 1 ˆS2 (See [Per02a, Per02b, Per03], [BBB `10] and [START_REF] Tian | Ricci flow and the Poincaré conjecture[END_REF]).

Because π 2 pM q and π 3 pM q are both trivial, then the spherical part of Σ and the 2spheres in Σ are homotopic to a point in M 4 . That is to say, Σ is homotopic to a wedge sum of some circles in M (these circles come from the S 2 ˆS1 's part of Σ 3 ). Because dimpM q " 4, we see that the intersection number of Σ and γ equals zero, which is in contradiction with the intersection number of γ and Σ.

Organization of the thesis

The plan of this thesis is as follows:

For the first part, we discuss contractible 3-manifolds and related topological properties. In Chapter 1, we recall related background from 3-manifolds, such as simplyconnectedness at infinity, the fundamental group at infinity and handlebodies. Subsequently, we discuss the topological structure of contractible 3-manifolds and derive some notations such as meridians of a handlebody, a system of meridians. Finally, we give some examples of 3-manifolds such as the Whitehead manifold and genus one 3-manifolds.

In Chapter 2, we start with the embedded discs in the Whitehead manifold. An interesting fact is that the behavior of these discs is influenced by the geometric index. Their relation is suggested by Theorem 2.1.2. Based on this relation, we introduce a new property, called Property P and we show that any contractible genus one 3-manifold satisfies this property (See Theorem 2.1.6).

Generally, a contractible 3-manifold may not satisfy Property P. The reason is that it may be made up of some handlebodies of higher genus. Some handles in these handlebodies may make no contribution to its topology and yield technical difficulties. To overcome it, we introduce two types of surgeries. Using these surgeries, we find a new family of handlebodies with good properties, called Property H (See Definition 2.2.5).

In the second part of the thesis, we focus on minimal surfaces and related convergence theories. In Chapter 3, we recall some notations such as the so-called first and second variation formulas, Morse index and the stable condition (See Chapter 3.1.1). Then we discuss Plateau's problem(See Chapter 3.1.2).

Subsequently, we focus on the local properties of minimal surfaces, including the strong maximal principle (See Corollary 3.2.3) and the monotonicity formula (See Proposition 3.2.5).

Then, we study the topology of stable minimal hypersurfaces. These hypersurfaces are characterized by the first eigenvalue of the stable operator (See Lemma 3.3.1 and Theorem 3.3.4). In a manifold of positive scalar curvature, there are many topological constraint for stable minimal surfaces. For example, if a complete 3-manifold has nonnegative scalar curvature, a complete stable minimal surface in it satisfies the extrinsic Cohn-Vossen inequality (See Corollary 3.3.6 and Theorem 3.3.10). As a consequence, we give a new proof of the topological classification of stable minimal surfaces in a 3-manifold with nonnegative scalar curvature (See Corollary 3.3.11 and [START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF]). Finally, as an application of minimal surfaces theory, we give the proof of Theorem A (See Theorem 3.3.12).

In Chapter 4, we discuss the convergence theory of minimal surfaces. We begin with the convergence of minimal surface equations (See Lemma 4.1.1). It can be generalized to the Riemannian case. Therefore we get a compactness theorem for minimal surfaces (See Theorem 4.1.4).

Next, we discuss the convergence without area estimate. In this case, the limit is a minimal lamination (i.e. a disjoint union of some embedded discs) instead of a single surface. We recall the minimal lamination theory of Colding-Minicozzi (See Appendix B of [START_REF] Colding | The space of embedded minimal surfaces of fixed genus in a 3-manifold. iv: Locally simply connected[END_REF]). Then we construct a required family of minimal laminations in a contractible 3-manifold. Their limit is a stable minimal lamination. Each leaf is a complete minimal surface. If the manifold has positive scalar curvature, it is a properly embedded plane (See Theorem D). As an application, we give a new proof of the topological classification of contractible 3-manifolds with uniformly positive scalar curvature (See Corollary 4.2.7 and [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF]).

For the third part, we give the complete proofs of the main theorems. In Chapter 5, we introduce the Vanishing property and study its relationship with the fundamental group at infinity, which is suggested by Lemma 5.2.1. Note that, in the genus one case, Property P implies the Vanishing property (See Theorem 5.1.1).

In Chapter 6, we reduce the proof of the main theorems to a cover lemma (See Lemma 6.1.3). For the proof of this lemma, we use the Vanishing property of the lamination (constructed in Chapter 4.2) to define a set S (See Definition 6.3.1). Positivity of the scalar curvature implies the finiteness of S (See Lemma 6.3.4 and Lemma 6.3.5). We use

Introduction (français)

La géométrie riemannienne vise à étudier les variétés riemanniennes qui sont des variétés lisses à structures métriques. Une des questions fondamentales est de comprendre la relation entre la courbure, définie localement, et les propriétés globales des variétés lisses. Le résultat le plus ancien est le théorème classique de Gauss et Bonnet, qui relie la courbure au nombre d'Euler, un invariant topologique. Ce théorème implique, par exemple, que toute surface compacte du genre g ą 0 n'a pas de métrique à courbure positive.

En dimension supérieure, l'existence de mtriques à courbure positive devient beaucoup plus compliquée parce qu'il existe plusieurs obstructions topologiques pour qu'une variété lisse ait une métrique complète à courbure positive.

Nous prenons ensuite 3-variétés pour expliquer ce fait.

pM 3 , gq cas compact cas non compact

K ą 0 S 3 {Γ R 3 Ric ą 0 R 3 Scal ą 0 p# k i"1 S 3 {Γ i q#p# l j"1 S 1 ˆS2 q ? Table 2.
Pour une 3-variété riemannienne compacte, il existe une solution unique en temps court au flot de Ricci (normalisé), introduit par Hamilton [START_REF] Hamilton | Three-manifolds with positive ricci curvature[END_REF]. Si la variété a une courbure de Ricci positive, la solution en temps court peut être étendue en la solution en temps long. La limite de ce flot est une métrique à courbure sectionnelle constante. Autrement dit, si une 3-variété compacte a courbure de Ricci positive, elle est homéomorphe au quotient S 3 {Γ de la sphère S 3 par un sous-groupe fini Γ Ă Op4q. Un tel quotient est appelé une 3-variété sphérique.

La contribution majeure suivante au sujet a été apportée par Perelman [Per02a, Per02b, Per03], qui a développé le flot de Ricci avec chirurgie. Un résultat surprenant et magnifique de cette étude est la preuve qu'une 3-variété compacte á courbure scalaire positive est homéomorphe à une somme connexe de certaines 3-variétés sphériques et de copies de S 1 ˆS2 (Voir aussi [BBB `10] et [START_REF] Tian | Ricci flow and the Poincaré conjecture[END_REF]). Sa généralisation au cas non compact est due à Bessières, Besson et Maillot [START_REF] Bessières | Ricci flow on open 3-manifolds and positive scalar curvature[END_REF].

Pour le cas non compact, les premiers résultats montrés par Gromoll et Meyer [START_REF] Gromoll | On complete open manifolds of positive curvatur[END_REF] sont qu'une 3-variété complète non compacte à courbure sectionnelle positive est homéomorphe à R 3 (Voir [START_REF] Gromoll | On complete open manifolds of positive curvatur[END_REF]).

L'étape suivante du développement de ce sujet est due à Schoen et Yau [START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF] qui ont utilisé la théorie des surfaces minimales et le théorème de splitting [START_REF] Cheeger | The splitting theorem for manifolds of nonnegative ricci curvature[END_REF] pour étudier la topologie des 3-variétés . Ils ont prouvé qu'une 3-variété complète non compacte à courbure de Ricci positive est homéomorphe à R 3 .

Bien que toutes ces travauw soient trés impressionnants, ils ont toujours laissé la ouverte question (voir le problème 27 dans [START_REF] Shing | Problem section[END_REF]):

Comment classifier les 3-variétés non compactes à courbure scalaire positive, à difféomorphisme près ?

Le but est de trouver toutes les obstructions et de caractriser toutes les 3-variètés ouvertes à courbure scalaire positive. Bien que Gromov-Lawson [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF] et Schoen-Yau [START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF] aient donné plusieurs obstructions topologiques, toutes ces classes d'obstruction disparaissent pour les 3-variétés contractiles.

Considérons des 3-variétés contractiles. Par exemple, R 3 possède une métrique complète g 1 à courbure scalaire positive, où

g 1 " 3 ÿ i"1 pdx i q 2 `p 3 ÿ i"1 x i dx i q 2 .
Jusqu'à présent, c'est la seule 3-variété contractile connue qui admet une métrique complète à courbure scalaire positive. Ceci suggère la question suivante :

Est-ce qu'une 3-variété complète contractile à courbure scalaire positive est homéomorphe à R 3 ? Une 3-variété complète contractile à courbure scalaire uniformément positive (c'està-dire que sa courbure scalaire est minorée par une constante strictement positive) est homéomorphe à R 3 . Cela a été prouvé pour la premire fois par Gromov et Lawson [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF]. Récemment, ce résultat a été généralisé par Chang, Weinberger et Yu [START_REF] Chang | Taming 3-manifolds using scalar curvature[END_REF] à des 3-variétés contractile dont la courbure scalaire est uniformément positive à l'extérieur d'un ensemble compact. En utilisant la théorie des surfaces minimales, nous généralisons davantage.

Théroème 1 (=Theorem A) Supposons que pM 3 , gq est une 3-variété complète contractile. S'il existe un réel α P p´8, 2q tel que lim inf rpxqÑ8 r α pxqκpxq ą 0, o κpxq est la courbure scalaire de pM, gq et rpxq est la fonction de distance d'un point 0 P M à x, alors M 3 est difféomorphe à R 3 . La preuve suit les arguments de Gromov et Lawson (voir le corollaire 10.9 dans [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF]).

3-variétés contractiles

La conjecture de géométrisation, formulée par William Thurston [Per02a, Per02b, Per03] (voir aussi [BBB `10] et [START_REF] Tian | Ricci flow and the Poincaré conjecture[END_REF]) énonce que pour la topologie d'une 3-variété compacte est entièrement déterminée par son type d'homotopie. Cependant, la structure topologique des 3-variétés contractiles est trés complique. Par exemple, la variété Whitehead (construite dans [START_REF] Whitehead | A certain open manifold whose group is unity[END_REF]) est une 3-variété contractile qui n'est pas homéomorphe à R 3 .

Pour expliquer la construction de la variété de Whitehead, introduisons le concept de méridien. Un méridien γ Ă BN d'un tore solide fermé N est une courbe fermée plongée qui est homotopiquement triviale dans N mais non contractile dans BN . Un disque méridien pD, BDq Ă pN, BN q d'un tore solide N est un disque plongé dont la frontière est un méridien de N . (Voir Définition 1.2.1).

La variété de Whitehead est construite à partir de l'entrelacs de Whitehead. Rappelons que l'entrelacs de Whitehead est un entrelacs á deux composantes comme illustr sur la figure suivante: Choisissons un tore solide fermé T 1 qui est non noué dans S 3 . L'intérieur du tore solide dans S 3 est un autre tore solide. Prenons un deuxième tore solide T 2 à l'intérieur de T 1 de sorte que l'intérieur de K 2 forme un entrelacs de Whitehead avec un méridien de T 1 comme sur la figure suivante. Le tore solide T 2 est non nou dans S 3 . Ensuite, on plonge T 3 dans T 2 de la même manière que T 2 se trouve dans T 1 et ainsi de suite un nombre infini de fois. Définissons l'ensemble T 8 " Ş 8 k"1 T k , appelé le continuum de Whitehead. La variété de Whitehead est définie comme suit: Wh :" S 3 zT 8 , qui est une 3-variété non compacte.

Remarque Puisque chaque T k est non noué dans S 3 , son complémentaire N k est un tore solide. Par conséquent, la variété de Whitehead est une union croissante de tores solides tN k u k . Chaque N k est plongé dans N k`1 de la même manière que T 2 dans T 1 . Cela découle de la symétrie de l'entrelacs de Whitehead.

Une variation de la construction, comme changer le noeud à chaque étape k, donne une famille de ce qu'on appelle les 3-variétés de genre un, introduite dans [START_REF] Dr Mcmillan | Some contractible open 3-manifolds[END_REF]. La construction est reliée à l'indice géométrique.

Si N 1 Ă N est un tore solide, l'indice géométrique, IpN 1 , N q, de N 1 dans N est égal au nombre minimal de points de l'intersection de l'intérieur de N 1 avec un disque mridien de N . Une 3-variété de genre un est une union croissante de tores solides tN k u telle que pour chaque k, N k Ă Int N k`1 , et que l'indice géométrique de N k dans N k`1 ne soit pas égal à zéro. (Voir Définition 1.3.8, Définition 1.3.12 et [START_REF] Garity | Contractible 3-manifolds and the double 3-space property[END_REF]).

Par exemple, Wh est une 3-variété contractile de genre un. L'indice géométrique IpN k , N k`1 q est égal à deux pour chaque k, où N k est illustr comme ci-dessus. Remarquons que R 3 n'est pas de genre un, mais de genre zéro, puisqu'il s'agit d'une union croissante de 0-corps à anses. (c'est-à-dire de 3-boules).

Une question intéressante est de savoir si la variété de Whitehead admet une métrique complète de courbure scalaire positive. Dans cette thèse, nous répondons négativement :

Théroème 2 (=Theorem B 1 ) La variété de Whitehead n'a pas de métrique complète á courbure scalaire positive.

Ce résultat peut être généralisé au cas du genre un.

Théroème 3 (=Theorem B 2 ) Une 3-variété contractile genre un n'a pas de métrique complète á courbure scalaire positive.

En combinant le travail de Kazdan [START_REF] Kazdan | Deformation to positive scalar curvature on complete manifolds[END_REF], nous généralisons ces résultats à la courbure scalaire non négative.

Corollarie (voir Corollaire 6.4.3)Une 3-variété contractile genre un n'a pas de métrique complète á courbure scalaire non négative L'existence de métriques complètes à courbure scalaire positive est liée au groupe fondamental à l'infini. Le groupe fondamental à l'infini π 8 1 d'une 3-variété est la limite projective des groupes fondamentaux des complémentaires de sous-ensembles compacts (voir Définition 1.1.9).

La trivialité du groupe fondamental à l'infini n'est pas équivalente pas à la connexité simple à l'infini. Par exemple, la variété de La raison en est la suivante : supposons qu'il existe une suite décroissante d'entiers tk n u et une suite tL tn u de feuilles telles que pour chaque n , L tn X BN kn a au moins un cercle non homotopiquement trivial dans BN kn . Comme pour la propriété de la suite tΩ k u k (Propriété P ), nous savons que le nombre de composantes connexes de L tn X N 1 intersectant N 0 tend vers l'infini quand k n tend vers l'infini (Voir Définition 2.1.3 et Théorème 2.1.6).

La suite tL tn u sous-converge vers une feuille L t inf ty dans la lamination L avec une multiplicit finie. La sous-convergence est assurée par un résultat de Schoen [START_REF] Schoen | Estimates for stable minimal surfaces in three dimensional manifolds[END_REF] (voir Lemme 4.1.5) et l'inégalité extrinsèque de Cohn-Vossen (voir Théorème 5). Par conséquent, L t8 X N 1 a une infinité de composantes intersectant N 0 . Comme dans le cas ci-dessus, chaque composante a une aide donnée. L'inégalité extrinsèque de Cohn-Vesson donne une contradiction. Dans le cas du genre un, la famille tN k u a plusieurs bonnes propriétés. Par exemple, les applications π 1 pBN k q Ñ i 1 pM zN k q et π 1 pBN k q Ñ π 1 pN k zN 0 q sont toutes les deux injectives (voir le Lemme 1.3.10). Ces propriétés sont cruciales et nécessaires dans l'étude de l'existence de métriques complètes à courbure scalaire positive. En général, la famille tN k u peut ne pas avoir les propriétés ci-dessus.

Par exemple, l'application π 1 pBN 0 q Ñ π 1 pM zN 0 q peut ne pas être injective. Pour surmonter cette difficulté, nous utilisons des chirurgies topologiques sur N 0 et nous trouvons un nouveau corps à anses pour le remplacer. Précisément, nous utilisons le lemme de la boucle pour trouver un disque plong pD, BDq Ă pM zN 0 , BN 0 q dont le bord est une courbe simple non contractile dans BN 0 . Le nouveau corps à anses est obtenu à partir de N 0 en attachant un voisinage tubulaire fermé N pDq de D dans M zN 0 .

Nous utilisons à plusieurs reprises des chirurgies topologiques sur chaque N k pour obtenir une nouvelle famille tR k u k de corps à anses fermés avec les propriétés suivantes, appelée Propriété H (voir Définition 2.2.5):

(1) l'application π 1 pBR k q Ñ π 1 pR k zR 0 q est injective pour k ą 0;

(2) l'application π 1 pBR k q Ñ π 1 pM zR k q est injective pour k ě 0;

(3) chaque R k est contractile dans R k`1 mais n'est pas contenu dans une 3-boule de M ; (4) il existe une suite d'entiers croissante tj k u k , telle que π 1 pBR k X BN j k q Ñ π 1 pBR k q soit surjective.

Remarque

avec BΩ l`1 k " γ l`1 k .
Considérons la variété riemannienne pT k,l , g| T k,l q, où T k,l :" N k z š l i"1 Ω l k . C'est un corps à anses du genre gpN k q ´l. Par exemple, voir la figure suivante. La suite tL tn u sous-converge vers une feuille de L . Pour simplifier, nous pouvons supposer que la suite tL tn u converge vers la feuille L t8 . La feuille L t inf ty satisfait la propriété d'annulation. C'est-à-dire qu'il existe un entier positif kpL t8 q tel que pour k ě kpL t8 q, tout cercle BR k X L t8 soit homotopiquement trivial dans BR k .

Ω 1 k pN k , g k q γ 1 k pT k,1 , g k | T k,1 q Ω 1 k γ1 k ´Ω1 k γ1 k Figure 0.6. La frontière de pT k,l , g| T k,l q est constituée de BN k z š l i"1 γ i k et certains disques disjoints tΩ i k ´ul i"1 et tΩ i k `ul i"1 . Les deux disques Ω i k ´et Ω i k `
Cependant, comme L tn XBR kn a un cercle non homotopiquement trivial dans partialR kn , nous savons que pour k n ą kpL t8 q, L tn X BR kpLt 8 q a un méridien de R kpLt 8 q (voir la remarque 2.2.7 et le corollaire 1.2.6). Ces méridiens de R kpLt 8 q convergeront vers un méridien de R kpLt 8 q contenu dans L t8 XBR kpLt 8 q . Ceci est en contradiction avec le dernier paragraphe.

Expliquons comment déduire une contradiction de la propriété d'annulation de L . Nous montrons que si N k contient R k 0 (pour k assez grand), alors L k X BR k 0 contient au moins un méridien de R k 0 (voir le Corollaire 1.2.8). Comme dans le cas ci-dessus, ces méridiens de R k 0 convergeront vers un cercle non contractile dans L X BR k 0 . La propriété d'annulation de L donne une contradiction.

4-variétés fermées asphériques

Une variété M est dite asphérique s'il est connexe par arcs et si tous ses groupes d'homotopie suérieurs s'annulent (c'est-à-dire que π k pM q est trivial pour k ě 2). La classe des variétés asphériques comprend toutes les variétés hyperboliques et toutes les variétés à courbure négative.

Une question intéressante est de savoir si le tore T n , n ě 3, possède une métrique à courbure scalaire positive. Cette question a été traitée par Gromov-Lawson [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF] et Schoen-Yau [START_REF] Schoen | On the structure of manifolds with positive scalar curvature[END_REF][START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF][START_REF] Schoen | Positive scalar curvature and minimal hypersurface singularities[END_REF]. En général, on conjecture que Conjecture. Aucune variété compacte asphérique n'a de métrique à courbure scalaire positive.

Cette conjecture a été démontrée pour les 3-variétés par Gromov et Lawson [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF]. En dimension quatre, elle est confirmée pour les 4-variété qui contiennent des surfaces incompressibles [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF].

Dans cette thèse, nous prouvons que

Théroème 6 (=Theorem F ) Aucune 4-variété asphérique fermée avec premier nombre de Betti non nul n'a de métrique à courbure scalaire positive.

Notons qu'il existe une 4-variété asphérique fermée dont le premier nombre de Betti est nul (Voir [START_REF] John | Some examples of aspherical 4-manifolds that are homology 4-spheres[END_REF]).

Raisonnons par l'absurde. Supposons qu'il existe une 4-variété asphrique fermée pM 4 , gq à courbure scalaire positive, dont le premier nombre de Betti b 1 pM 4 q est strictement supérieur à zéro.

Choisissons un cercle γ Ă M 4 tel que rγs soit d'ordre infini dans H 1 pM 4 , Zq. Nous utilisons la dualité de Poincaré pour trouver une classe u P H 3 pM 4 , Zq avec ă u, r gs ą" 1. Un théorème de Fleming-Federer (voir [START_REF] Federer | Normal and integral currents[END_REF] ou le chapitre 7 de [START_REF] Simon | Lectures on geometric measure theory[END_REF]) nous indique qu'il existe une hypersurface Σ 3 minimisant le volume dans cette classe. Par conséquent, le nombre d'intersection de γ et Σ 3 est égal à un.

Puisque pM, gq a une courbure scalaire positive, alors Σ 3 admet une métrique de courbure scalaire positive (voir la proposition 3.3.5). La variété Σ 3 est homéomorphe à une somme connexe de 3-variété sphériques et de copies de

S 1 ˆS2 (Voir [Per02a,Per02b, Per03], [BBB `10] et [MT07]).
Comme π 2 pM q et π 3 pM q sont triviaux, la partie sphérique de Σ et les 2-sphères de Σ sont homotopes à un point dans M 4 . C'est-à-dire que Σ est homotope à un bouquet de cercles dans M (ces cercles proviennent de la partie de S 2 ˆS1 dans Σ 3 ). Comme dimpM q " 4, nous voyons que le nombre d'intersection de Σ et γ est égal à zéro, ce qui est en contradiction avec les nombres d'intersection de γ et Σ.

Organisation de la thèse

Le plan de cette thèse est le suivant: Dans la première partie, nous discutons des 3-variétés contractile et des propriétés topologiques associées. Au Chapitre 1, nous formulons des rappels liés aux 3-variétés, comme la connexité simple à l'infini, le groupe fondamental à l'infini et les corps à anses. Par la suite, nous discutons de la structure topologique des 3-variétés contractiles et présentons quelques notions telles que les méridiens d'un corps à anses, ou encore les systèmes de méridiens. Enfin, nous donnons quelques exemples de 3-variétés, telles que les variétés de Whitehead et les 3-variétés de genre un.

Au chapitre 2, nous commenons par les disques plongés dans la variété de Whitehead. Un fait intéressant est que le comportement de ces disques est influencé par l'indice géométrique. Leur relation est suggérée par le Théoréme 2.1.2. Sur la base de cette relation, nous introduisons une nouvelle propriété, appelée Propriété P, et montrons que toute variété contractile de genre un satisfait cette propriété (voir Théorème 2.1.6).

En règle générale, une 3-variété contractible peut ne pas satisfaire la propriété P. La raison est que cette variété peut être composée de corps à anses de genre supérieur. Certaines anses peuvent ne pas contribuer à la topologie et engendrer des difficultés techniques. Pour surmonter cela, nous introduisons deux types de chirurgies. En utilisant ces chirurgies, nous trouvons une nouvelle famille de corps à anses avec de bonnes propriétés, appelée Proprit H (voir Définition 2.2.5).

Dans la deuxième partie de la thèse, nous nous concentrons sur les surfaces minimales et les théories de convergence associées. Au Chapitre 3, nous rappelons certaines notions telles que les formules dites de première et deuxième variation, l'indice de Morse et la condition de stabilité (Voir Chapitre 3.1.1). Ensuite, nous discutons du problème de Plateau (voir Chapitre 3.1.2).

Nous nous intéresserons ensuite aux propriétés locales des surfaces minimales, y compris le principe du maximum et la formule de monotonie (voir Proposition 3.2.5).

Ensuite, nous étudions la topologie des hypersurfaces minimales stables. Ces hypersurfaces sont caractérisées par la première valeur propre de l'opérateur stable (voir le Lemme 3.3.1 et le Théorème 3.3.4). Dans une variété à courbure scalaire positive, il existe de nombreuses contraintes topologiques sur les surfaces minimales stables. Par exemple, si une 3-variété compléte a courbure scalaire positive ou nulle, une surface minimale stable complète vérifie l'inégalit extrinsèque de Cohn-Vossen (Voir Corollary 3.3.6 et Theorem 3.3.10). En conséquence, nous donnons une nouvelle preuve de la classification topologique des surfaces minimales stables dans une 3-variété à courbure scalaire positive ou nulle (voir le Corollaire 3.3.11 et [START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF]). Enfin, en tant qu'application de la théorie des surfaces minimales, nous donnons la preuve du Théorème 4.2.1 (voir Théorème 3.3.12).

Au Chapitre 4, nous discutons de la théorie de la convergence des surfaces minimales. Nous commenons par la convergence des équations de surfaces minimales (Voir Lemme 4.1.1). Cela peut être généralisé au cas riemannien. Par conséquent, nous obtenons un théorème de compacité pour les surfaces minimales (voir Théorème 4.1.4).

Ensuite, nous discutons de la convergence sans estimation d'aire. Dans ce cas, la limite est une lamination minimale (c'est-à-dire une union disjointe de disques plongés). Nous rappelons la thorie de la lamination minimale de Colding-Minicozzi (Voir l'Annexe B de [START_REF] Colding | The space of embedded minimal surfaces of fixed genus in a 3-manifold. iv: Locally simply connected[END_REF]). Ensuite, nous construisons une famille requise de laminations minimales dans une 3-variété contractile. Leur limite est une lamination minimale stable. Chaque feuille est une surface minimale compléte. Si la variété a une courbure scalaire positive, il s'agit d'un plan proprement plongé (voir le Théorème 5). En guise d'application, nous donnons une nouvelle preuve de la classification topologique des 3-variétés contractiles à courbure scalaire uniformément positive (Voir Corollaire 4.2.7 et [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF]).

Dans la troisième partie, nous donnons les preuves complètes des théorèmes principaux. Au Chapitre 5, nous introduisons la propriété d'annulation et étudions sa relation avec le groupe fondamental à l'infini. La relation est suggérée par le Lemme 5.2.1. Notez que, dans le cas du genre un, la propriété P implique la propriété d'annulation (voir Théorème 5.1.1).

Au Chapitre 6, nous réduisons la preuve des théorèmes principaux à un lemme de recouvrement (voir Lemme 6.1.3). Pour prouver ce lemme, nous utilisons la propriété d'annulation de la lamination (construit au Chapitre 4.2) pour définir un ensemble S (voir Définition 6.3.1). La positivité de la courbure scalaire implique la finitude de S (voir Lemme 6.3.4 et Lemme 6.3.5). Nous utilisons la finitude pour prouver le lemme de recouvrement. Enfin, nous discutons de plusieurs questions connexes et d'autres sujets de recherche sur les 3-variétés.

Dans la quatrième partie de la thèse, nous discutons de l'existence de métriques à courbure scalaire positive sur une 4-variété asphérique compacte. Avec un résultat de Perelman [Per02a, Per02b, Per03], nous donnons une preuve du Théorème 6. Enfin, nous abordons plusieurs autres questions sur les 4-variétés.

Part 1

3-Manifolds

CHAPTER 1

3-Manifolds

In this chapter, we review related background in geometric topology and algebraic topology.

We begin with several classical theorems in 3-manifolds, such as the loop lemma. Subsequently, we study the topological structures of contractible 3-manifolds. Then we introduce several notations (for example, effective meridians and a system of meridians) and discuss their topological properties.

Finally, we give several examples, such as genus one 3-manifolds.

Background

In this part, we recall several classical theorems for 3-manifolds and discuss the topological structure of contractible 3-manifolds.

1.1.1. Preliminary. A 3-manifold is irreducible if any embedded 2-sphere bounds a closed 3-ball (namely, it is homeomorphic to a closed unit ball in R 3 ).

Remark 1.1.1. We know from the so-called Alexander's theorem (See [Theorem 1.1, Page 1] of [START_REF] Hatcher | Notes on basic 3-manifold topology[END_REF]) that any embedded 2-sphere in R 3 bounds an embedded 3-ball.

Further, the proof of Poincaré Conjecture [Per02a, Per02b, Per03] (See [BBB `10] or [START_REF] Tian | Ricci flow and the Poincaré conjecture[END_REF]) tells that any contractible 3-manifold is irreducible.

It is well-known that there are many links between the geometric properties of 3manifolds and homotopy theory, specially π 1 . For example, the loop lemma.

Lemma 1.1.2. (See [Theorem 3.1, Page 54] of [START_REF] Hatcher | Notes on basic 3-manifold topology[END_REF])Let M be a 3-manifold with boundary BM , not necessarily compact or orientable. If there is a map f : pD 2 , BD 2 q Ñ pM, BM q with the property that f | BD 2 is not nullhomotopic in BM . Then there is an embedding h with the same property.

Remark 1.1.3. We may assume that hpInt D 2 q Ă Int M . The reason is described below:

Let us consider a 1-sided open neighborhood M -BM ˆr0, q of BM in M . Shrinking the image of f into M p q :" M zM , we find a map f : pD 2 , BD 2 q Ñ pM p q, BM p qq with the property that f pBD 2 q is not nullhomotopic in BM p q. By Lemma 1.1.2 there is an embedding h with the same property. Its image is contained in pM p q, BM p qq. Therefore, the image of h is contained in Int M .

In addition, there is an embedded circle γ Ă BM which is homotopic to h pBD 2 q in M . There is an embedded annulus A Ă M joining γ and h pBD 2 q. We have a map h : pD 2 , BD 2 q Ñ pM, BM q so that its image is an embedded disc (i.e. the union of A and the image of h ). It has the same property as f and hpInt D 2 q Ă Int M .

One of tools for computing the fundamental group is the so-called Van-Kampen's theorem. It will be frequently used in the following.

Theorem 1.1.4. (Van-Kampen's Theorem, See [Theorem11.60, Page 396] of [START_REF] Joseph J Rotman | An introduction to the theory of groups[END_REF]) Let K be a connected complex having connected subcomplexes L 1 and L 2 with K " L 1 YL 2 . If L 1 X L 2 is connected, then π 1 pKq is the pushout of the data.

π 1 pL 1 q / / π 1 pKq π 1 pL 1 X L 2 q j 1˚O O j 2˚/ / π 1 pL 2 q O O where j k : L 1 X L 2 Ñ L k is the inclusion for k " 1, 2. Moreover, π 1 pKq is isomorphic to π 1 pL 1 q ˚π1 pL 1 XL 2 q π 1 pL 2 q.
If the map j k ˚: π 1 pL 1 X L 2 q Ñ π 1 pL k q is injective for k " 1, 2, π 1 pKq can be written as a free product with amalgamation (See Chapter 11 of [START_REF] Joseph J Rotman | An introduction to the theory of groups[END_REF]).

Lemma 1.1.5. (See [Theorem11.67, Page 404] of [START_REF] Joseph J Rotman | An introduction to the theory of groups[END_REF]) Let B, A 1 and A 2 be groups. . Let A 1 ˚B A 2 be the pushout of the following data.

A 1 λ 1 / / A 1 ˚B A 2 B i 1 O O i 2 / / A 2 λ 2 O O If the map i k : B Ñ A k is injective for k " 1, 2, one has (1) the map λ k is injective for k " 1, 2; (2) if A 1 k " λ k pA k q, then ă A 1 1 , A 1 2 ą" A 1 ˚B A 2 and A 1 1 X A 1 2 is isomorphic to B.
We now introduce several concepts about the disjoint closed curves in a disc. Definition 1.1.6. (See Definition 2.11 of [START_REF] Wang | Contractible 3-manifolds and positive scalar curvatures (I)[END_REF]) Let C :" tc i u iPI be a finite set of pairwise disjoint circles in the disc D 2 and D i Ă D 2 the unique disc with boundary c i . Consider the set tD i u iPI and define the partially ordered relation induced by the inclusion. For each maximal element D j in ptD i u iPI , Ăq, its boundary c j is defined as a maximal circle in C. For each minimal element D j , its boundary c j is called a minimal circle in C.

Simply-connectedness at infinity and π 8

1 .

Definition 1.1.7. A topological space M is simply connected at infinity if for any compact set K Ă M , there exists a compact set K 1 containing K so that the induced map π 1 pM zK 1 q Ñ π 1 pM zKq is trivial.

A result of Stallings [Sta72] and Remark 1.1.1 tell us that the only contractible and simple-connected at infinity 3-manifold is R 3 .

Remark 1.1.8. If a contractible 3-manifold M is not homeomorphic to R 3 , it is not simply-connected at infinity. That is to say, there is a compact set K Ă M so that for any compact set K 1 Ă M containing K, the induced map π 1 pM zK 1 q Ñ π 1 pM zKq is not trivial. We also have that the set K is not contained in a 3-ball in M . The reason is described below:

If a closed 3-ball B contains K, Theorem 1.1.4 shows that π 1 pM q -π 1 pM zBq ˚π1 pBBq π 1 pBq. In addition, π 1 pBq and π 1 pBBq are both trivial. Therefore, π 1 pM zBq -π 1 pM q is trivial. That is to say, the map π 1 pM zBq Ñ π 1 pM zKq is trivial. This is a contradiction. Definition 1.1.9. The fundamental group at infinity π 8 1 of a path-connected space is the inverse limit of the fundamental groups of complements of compact subsets.

For example, the fundamental group at infinity of any compact manifold is trivial. For any contractible n-manifold M n , it is simply-connected at infinity if and only if π 8 1 pM n q is trivial, when n ě 4 (See [START_REF] Chang | Taming 3-manifolds using scalar curvature[END_REF]). However, this result is not true in dimension 3. For example, the Whitehead manifold is not simply-connected at infinity but its fundamental group at infinity is trivial.

Remark 1.1.10. Let us consider a contractible 3-manifold M and an exhaustion tN k u k of M . We have the following:

π 8 1 pM q " lim ÐÝ k π 1 pM zN k q " # prγ k sq P ź k π 1 pM zN k q so that pf k,j q ˚prγ k sq " rγ j s for any k ě j + ,
where f k,j is the inclusion from M zN k to M zN j .

Therefore, π 8 1 pM q is non-trivial if and only if there exists a non-trivial element prγ k sq P ś k π 1 pM zN k q satisfying 1) for some k 0 , the closed curve

γ k 0 is non-contractible in M zN k 0 ; 2) for k ě k 0 , γ k is homotopic to γ k in M zN k 0 .
That is to say, there is a compact set K and a family of closed curves tα n u n going to infinity with the following property: for each n a) α n is nullhomotopic in M zK for ; b) α n is homotopic to α n`1 in M zK. Note that this family of circles gives a non-trivial element in π 8 1 pM q. 1.1.3. Handlebodies.

Definition 1.1.11. [Page 59, [START_REF] Rolfsen | Knots and links[END_REF]] A handlebody is any space obtained from the 3-ball D 3 (0-handle) by attaching g distinct copies of D 2 ˆr´1, 1s (1-handle) with the homeomorphisms identifying the 2g discs D 2 ˆt˘1u to 2g disjoint 2-disks on BD 3 , all to be done in such a way that the resulting 3-manifold is orientable. The integer g is called the genus of the handlebody.

Remark that a handlebody of genus g is homeomorphic to a boundary connected sum of g solid tori. Therefore, its boundary is a compact surface of genus g. (See Page 59 in [START_REF] Rolfsen | Knots and links[END_REF])

From a result of McMillan [START_REF] Dr Mcmillan | Cartesian products of contractible open manifolds[END_REF] and Remark 1.1.1, we know that:

Theorem 1.1.12. [Page 511, Theorem 1] [MJ61] Any contractible 3-manifold can be written as an ascending union of handlebodies .

Remark 1.1.13. Let consider a contractible 3-manifold M . If it is not homeomorphic to R 3 , it can written as an increasing family of handlebodies tN k u satisfying that for each k,

' N k is homotopically trivial in N k`1 ; ' none of the N k is contained in a 3-ball (See Remark 1.1.8).
In the following, we consider a closed handlebody N in S 3 .

Definition 1.1.14. A closed handlebody N Ă S 3 of genus g is said to be unknotted in S 3 if it complement in S 3 is also a handlebody of genus g.

For example, an unknotted solid torus in S 3 .

Meridians

In the following, we consider a closed handlebody N .

Definition 1.2.1. An embedded circle γ Ă BN is called a meridian if γ is nullhomot- pic in N , but not contractible in BN .
An embedded closed disc pD, BDq Ă pN, BN q is called a meridian disc if its boundary is a meridian of N .

The disc D is a splitting meridian disc, if N zD is not connected. Its boundary is called a splitting meridian.

The disc D is a non-splitting disc, if N zD is connected. Its boundary is called a non-splitting meridian.

Remark. Let γ be a meridian of N . If γ is a splitting meridian, it cuts BN into two components. The class rγs is equal to zero in H 1 pBN q.

If γ is a non-splitting meridian, then BN zγ is connected. The class rγs is a non-trivial element in H 1 pBN q.

Lemma 1.2.2. Let N 1 and N be two closed handlebodies with N 1 Ă Int N . If N 1 is homotopically trivial in N , then any non-splitting meridian of N 1 is non-trivial in H 1 pN zN 1 q and any meridian of N is trivial in H 1 pN zN 1 q.

Proof. The Mayer-Vietoris sequence gives:

(1.2.1)

H 2 pN q Ñ H 1 pBN 1 q i 1 Ñ H 1 pN 1 q ' H 1 pN zN 1 q i 2 Ñ H 1 pN q Ñ Ĥ0 pBN 1 q.
We know that H 2 pN q and Ĥ0 pBN 1 q are both trivial. Then, the map H 1 pBN 1 q Ñ H 1 pN 1 q ' H 1 pN zN 1 q is injective. As indicated above, any non-splitting meridian of N 1 is non-trivial in H 1 pBN 1 q. Therefore, it is not equal to zero in H 1 pN zN 1 q.

In the following, we will show that any meridian γ of N is trivial in H 1 pN zN 1 q.

Embed N into S 3 as an unknotted handlebody. The set N 1 can be viewed as a handlebody in S 3 . The core K 1 " _ g 1 i"1 α i of N 1 is a wedge sum of g 1 circles tα i u i , where g 1 is the genus of N 1 .

Choose γ 1 i Ă BN 1 as a normal circle of α i in S 3 . That is to say, the linking number of γ 1 i > α j in S 3 is equal to δ ij . In addition, the kernel of the map H 1 pBN 1 q Ñ H 1 pN 1 q is a linear subspace of dimension g 1 , spanned by trγ 1 i su g 1 i"1 .

Since N 1 is homotopically trivial in N , the map H 1 pN 1 q Ñ H 1 pN q is a zero map. Choose the element p0, rγsq P H 1 pN 1 q ' H 1 pN zN 1 q. Since γ is a meridian of N , the element is in the kernel of the map i 2 . From the sequence (1.2.1), it is also contained in the image of i 1 . Hence, rγs can be written as

ř i n i rγ 1 i s in H 1 pN zN 1 q. Claim: the coefficient n i is equal to the linking number of α i > γ.
From the Mayer-Vietoris sequence, the group H 1 pS 3 zK 1 q is a free Abelian group of rank g 1 spanned by trγ 1 i su g 1 i"1 . Hence, rγs is equal to

ř i n i rγ 1 i s in H 1 pS 3 zK 1 q.
Similarly, H 1 pS 3 zα i q is of rank one and generated by γ 1 i . One has that

H 1 pN zN 1 q Ñ H 1 pS 3 zK 1 q Ñ H 1 pS 3 zα i q rγs " ÿ i n i rγ 1 i s Þ Ñ ÿ i n i rγ 1 i s Þ Ñ n i rγ 1 i s That is to say, rγs is equal to n i rγ 1 i s in H 1 pS 3 zα i q.
From the definition of the linking number (See Page 132 in [START_REF] Rolfsen | Knots and links[END_REF]), n i is the linking number of γ > α i .

Since each α i Ă N 1 is nullhomotopic in N , then the linking number of γ > α i is zero. Namely, n i is equal to zero, for each i. Therefore, rγs is equal to zero in H 1 pN zN 1 q.

1.2.1. The effective meridian. Consider two closed handlebodies N 1 and N with

N 1 Ă Int N . Definition 1.2.3. A meridian γ of N is called an effective meridian relative to N 1 if any meridian disc with boundary γ intersects the core of N 1 .
The handbody N is called an effective handlebody relative to N 1 , if any meridian of N is an effective meridian relative to N 1 . Note that if N 1 is contained in a 3-ball B Ă Int N , there is no effective meridian relative to N 1 . Lemma 1.2.4. Let N 1 and N be two closed handlebodies with N 1 Ă Int N . The handlebody N is an effective handlebody relative to N 1 if and only if the map π 1 pBN q Ñ π 1 pN zN 1 q is injective.

Proof. If N is not an effective handlebody relative to N 1 , there is a meridian disc pD, BDq Ă pN, BN q with D X N 1 " H. Therefore, the map π 1 pBN q Ñ π 1 pN zN 1 q is not injective.

If the map π 1 pBN q Ñ π 1 pN zN 1 q is not injective, we apply Lemma 1.1.2 to the 3manifold N zN 1 . There is an embedded disc pD 1 , BD 1 q Ă pN zN 1 , BN q whose boundary is not contractible in BN . As in Remark 1.1.3, we may assume that Int D 1 Ă Int pN zN 1 q. We see that D 1 is a meridian disc with D 1 X N 1 " H. Therefore, N is not an effective handlebody relative to N 1 . Lemma 1.2.5. Let N 1 and N be two closed handlebodies satisfying that 1) N 1 Ă Int N and 2) π 1 pBN 1 q Ñ π 1 pN zN 1 q is injective. If N is an effective handlebody relative to N 1 , then any meridian disc pD, BDq Ă pN, BN q contains a meridian of N 1 .

The proof is the same as the proof of Lemma 2.12 in [START_REF] Wang | Contractible 3-manifolds and positive scalar curvatures (I)[END_REF].

Proof. Suppose that the closed meridian disc D intersects BN 1 transversally where γ :" BD is a meridian of N . The intersection D X BN 1 is a disjoint union of circles tc i u iPI . Each c i bounds a unique closed disc D i Ă Int D.

Consider the set C non :" tc i | c i is not contractible in BN 1 u and the set C max " tc i | c i is a maximal circle in tc i u iPI u.

We will show that C non is nonempty and a minimal circle in C non is a desired meridian. Suppose the contrary that C non is empty. Hence, each c i P C max is contractible in BN 1 and bounds a disc

D 1 i Ă BN 1 . Consider the immersed disc D :" pDz Y c i PC max D i q Y pY c i PC max D 1 i q with boundary γ. Since D X Int N 1 " H, we see that γ is contractible in N zN 1 .
However, Lemma 1.2.4 shows that the map π 1 pBN q Ñ π 1 pN zN 1 q is injective. That is to say, the circle γ is nullhomotopic in BN . This is in contradiction with our hypothesis that γ is non-trivial in π 1 pBN q. We conclude that C non ‰ H.

In the following, we will prove that each minimal circle c j in C non is a required meridian. From Definition 1.2.1, it is sufficient to show that c j is homotopically trivial in N 1 . Our strategy is to construct an immersed disc Dj Ă N 1 with boundary c j .

Let C j :" tc i |c i Ă Int D j for i P Iu and C max j be the set of maximal circles in C j . We now have two cases: C j " H or C j ‰ H.

Case I: If C j is empty, we consider the set Z :" Int D j and define the disc Dj as Int D j .

Case II: If C j is not empty, then C max j is also nonempty. From the minimality of c j in C non , each c i P C max j is nullhomotopic in BN 1 and bounds a disc D 2 i Ă BN 1 . Define the set Z :" Int D j z Y c i PC max j D i and the new disc Dj :" Z Y pY c i PC max j D 2 i q with boundary c j .

Let us explain why Dj is contained in N 1 . In any case, BN 1 cuts N into two connected components, N zN 1 and Int N 1 . The set Z is one of these components of Int D j zBN 1 . Therefore, it must be contained in Int N 1 or N zN 1 .

If Z is in N zN 1 , the disc Dj is contained in N zN 1 . Thus, c j is contractible in N zN 1 . However, since the induced map π 1 pBN 1 q Ñ π 1 pN zN 1 q is injective, then c j is homotopically trivial in BN 1 . This contradicts the choice of c j P C non . We conclude that Z is contained in Int N 1 . Therefore, Dj is contained in N 1 . That is to say, c j is null-homotopic in N 1 . However, rc j s is a non-trivial element in π 1 pBN 1 q. From Definition 1.2.1, we conclude that c j Ă D is a meridian of N 1 . This finishes the proof.

As a consequence, we have Corollary 1.2.6. Let N 1 and N be two closed handlebodies in a contractible 3manifold M satisfying that 1) N 1 Ă Int N and 2) the map π 1 pBN 1 q Ñ π 1 pM zN 1 q is injective. If an embedded circle γ Ă BN is not nullhomotopic in M zN 1 , then any embedded disc D Ă M with boundary γ contains a meridian of N 1 .

The proof is the same as Lemma 1.2.5.

The system of meridians.

Lemma 1.2.7. For a closed handlebody N of genus g, there are g disjoint non-splitting meridians tγ l u g l"1 so that N z > l N l pD l q is a closed 3-ball, where D l is a closed meridian disc with boundary γ l and N l pD l q is an open neighborhood of D l in N with small radius l .

The set of these meridians tγ l u g l"1 is called a system of the handlebody N of genus g. In general, it is not unique.

Proof. Pick any non-splitting meridian γ 1 of N . We use Lemma 1.1.2 to find an embedded disc D 1 Ă N .

As Remark 1.1.3, we may assume that Int D 1 Ă Int N . The set N 1 :" N zN pD 1 q is a closed handlebody of genus g ´1, where N 1 pD 1 q is the open tubular neighborhood of D 1 in N with small radius 1 . In particular, the map π 1 pBN X BN 1 q Ñ π 1 pBN 1 q is surjective.

Choose a non-splitting meridian γ 2 Ă BN X BN 1 of N 1 . By Lemma 1.1.2, there exists a meridian disc D 2 of N 1 " N zN 1 pD 1 q. The set N 2 :" N zN 1 pD 1 q > N 2 pD 2 q is a closed handlebody of genus g ´2, where N 2 pD 2 q is an open tubular neighborhood of D 2 in N .

We repeat this process g ´2 times and obtain g disjointly embedded discs tD l u so that N z > l N l pD l q is a handlebody of genus zero (a 3-ball). The boundaries tγ l u g l"1 of these discs are g distinct meridians which are the required candidates in the assertion.

Corollary 1.2.8. Let N Ă M , tγ l u and tD l u be as in Lemma 1.2.7, where M is a 3-manifold without boundary. If R Ă Int N is a closed handlebody satisfying that 1) it is not contained in a 3-ball in M ; 2) π 1 pBRq Ñ π 1 pM zRq is injective, then BR X > l D l contains at least a meridian of R.

The poof is also similar to the proof of Lemma 2.12 [START_REF] Wang | Contractible 3-manifolds and positive scalar curvatures (I)[END_REF].

Proof. We may assume that BR intersects > l D l transversally. The intersection BR X >D l :" tγu γPC has finitely many components. Let us consider the set C non :" tγ P C is not contractible in BRu.

Claim: C non is nonempty.

We argue by contradiction. Suppose that C non is empty. We have that any circle in D l X BR is contractible in BR. As in the proof of Lemma 1.2.5, we get a new disc in N zR with boundary γ l . Therefore, γ l is null-homotopic in N zR.

We use Lemma 1.1.2 to find an embedded disc

D 1 1 Ă N zR with boundary γ 1 . As in Remark 1.1.3, we may assume that Int D 1 1 Ă Int N zR (or D 1 1 Ă N zR). Choose the open tubular neighborhood N 1 1 pD 1 1 q of D 1 1 in N zR with small radius 1 1 . The set N 1 1 :" N zN 1 1 pD 1 1 q is a closed handlebody of genus g ´1 containing R. In addition, for l ą 1, γ l is a non-splitting meridian of N 1 1 but contractible in N zpN 1 1 pD 1 1 q > Rq.
Repeating this process g ´1 times, we obtain g embedded discs tD 1 l u g l"1 so that 1) R X > l N 1 l pD 1 l q " H; 2) The handlebody N z > l N 1 l pD 1 l q is of genus zero (a closed 3-ball), where N 1 l pD 1 l q is the open tubular neighborhood of D 1 l in N with small radius 1 l .

Therefore, R is contained in the 3-ball N z > l N 1 l pD 1 l q. This contradicts our hypothesis. The claim follows.

As in the proof of Lemma 1.2.5, we use the condition 2) to show that each minimal circle in C non is a required meridian.

Examples

In this part, we begin with Knot theory in a closed solid torus. Subsequently, we introduce several notations, such as the geometric index, the Whitehead manifold and contractible genus one 3-manifolds. In addition, we construct two non-homeomorphic contractible 3-manifolds whose fundamental group at infinity are both non-trivial.

1.3.1. Knots basic. Definition 1.3.1. A subset K of a 3-manifold X is a knot if K is homeomorphic with a circle S 1 . More generally, K is a link if K is homeomorphic
with a disjoint union of one or some circle(s).

Two knots or links K and K 1 are ambient isotopic if there is a homeomorphism h : X Ñ X such that (1) h is isotopic to the identity map; (2) hpKq " K 1 .

A knot K is called to be trivial (or unknotted ) in X if there exists an embedded disc in X with boundary K.

(II) (I) Figure 1.1.
For example, if X is R 3 or S 3 , the knot (I) is ambient isotopic to the knot (II) in X (See Figure 1.1). These two knots are both trivial in S 3 .

If X is a torus T 2 , two knots K and K 1 are ambient isotopic if and only if rKs " ˘rK 1 s in π 1 pT 2 q (See [16.Theorem, Page 25] of [START_REF] Rolfsen | Knots and links[END_REF]).

We consider a closed solid torus N and a knot K Ă N . Embed N into S 3 as an unknotted solid torus (See Definition 1.1.14). The knot K can be viewed as a knot in S 3 . The disjoint union K > γ is a link in S 3 , where γ is a meridian of N .

Recall that an n-component link L Ă S 3 is unlinked if and only if there exist n disjointly embedded discs D i Ă S 3 so that L " > i BD i .

Remark 1.3.2. The knot K is trivial in N if and only if the link K > γ is unlinked in S 3 . The reason is as follows:

If K is trivial in N , there exists an embedded closed disc D Ă N with boundary K. This disc is away from the knot γ in S 3 .

Since N Ă S 3 is unknotted , the meridian γ is a trivial knot in S 3 . Note that S 3 zD is homeomorphic to an open 3-ball. Then, γ is a trivial knot in S 3 zD. Hence, one finds an embedded disc D 0 Ă S 3 zD with boundary γ. Therefore, K > γ is the boundary of D > D 0 . That is to say, the link γ > K Ă S 3 is unlinked.

If K > γ is unlinked in S 3 , there exists an embedded closed disc D in S 3 zγ with boundary K. Therefore, K is trivial in the complement of the knot γ.

A closed solid torus N is homeomorphic with S 1 ˆD2 , where D 2 is a closed unit disc in R 2 . A special homeomorphism h : S 1 ˆD2 Ñ N is called a framing of N .
A longitude of N is any simple closed curve in BN of form hpS 1 ˆx0 q, for some framing h of N and some point x 0 in D 2 .

Remark 1.3.3. In a closed solid torus N , the kernel of the induced map π 1 pBN q Ñ π 1 pN q is isomorphic to Z. Each meridian γ of N belongs to the kernel. Since γ is an embedded curve, it is a generator of the kernel.

An embedded circle θ Ă N is a longitude if and only if rθs and rγs generate H 1 pBN, Zq, where γ is a meridian of N (See Page 29 of [START_REF] Rolfsen | Knots and links[END_REF]). In addition, any longitude of N is isotopic to the core of N in N .

Definition 1.3.4. Assume that p, q are relatively prime and N is an unknotted solid torus in S 3 . The torus knot T p,q Ă BN Ă S 3 of type pp, qq is the knot which wraps around N in the longitudinal direction p times and in the meridional direction q times. For example, the trefoil is T 2,3 . Here are T 2,3 and T 3,4 .

The p2, 3q torus knot The p3, 4q torus knot Remark 1.3.5. The knot T p,q is trivial in S 3 if and only if p " ˘1 or q " ˘1 (See Page 53 of [START_REF] Rolfsen | Knots and links[END_REF]).

In the following, we consider two closed solid tori N 1 and N with N 1 Ă IntN .

Lemma 1.3.6. If the closed solid torus N 1 Ă Int N is homotopically trivial in the closed solid torus N , then H 1 pN zN 1 q " Z 2 and the kernel of the induced map H 1 pBN 1 q Ñ H 1 pN zN 1 q is generated by a longitude of N 1 .

Proof. As in Lemma 1.2.2, we use the Mayer-Vietoris sequence to show that H 1 pN zN 1 q is isomorphic to Z 2 . It is generated by a meridian of N 1 and a longitude of N .

The image of the map H 1 pBN 1 q Ñ H 1 pN zN 1 q is a subgroup of rank one which is generated by the meridian γ 1 of N 1 . The kernel of H 1 pBN 1 q Ñ H 1 pN zN 1 q is also of rank one and generated by rθ 1 s, where θ 1 Ă BN 1 is an embedded circle. Therefore, H 1 pBN 1 q is generated by rγ 1 s and rθ 1 s. The circle θ 1 is a longitude of N 1 (See Page 29 of [START_REF] Rolfsen | Knots and links[END_REF] or Remark 1.3.3). That is to say, the longitude θ 1 is a generator of the kernel of H 1 pBN 1 q Ñ H 1 pN zN 1 q. 1.3.2. The Whitehead manifold. The Whitehead manifold is constructed from the Whitehead link. Recall that the Whitehead link is a link with two components illustrated in Figure 1.3: Choose a closed unknotted solid torus T 1 in S 3 . Its complement inside S 3 is another solid torus. Take a second solid torus T 2 inside T 1 so that the core of K 2 forms a Whitehead link with any meridian of T 1 as in Figure 1.4. The solid torus T 2 is unknotted in S 3 . Then, embed T 3 inside T 2 in the same way as T 2 lies in T 1 and so on infinitely many times. Define the set T 8 " X 8

k"1 T k , called the Whitehead continuum.

The Whitehead manifold is defined as W h :" S 3 zT 8 which is an open 3-manifold.

Remark 1.3.7. From the above construction, we know that (1) Since each T k is unknotted in S 3 , then its complement N k is a solid torus. Therefore, the Whitehead manifold is an increasing union of solid tori tN k u k as in Remark 1.1.13. In addition, each N k is embedded inside N k`1 in the same way as T 2 lies in T 1 . This follows from the symmetry of the Whitehead link.

(2) The core K k of N k is a non-trivial knot in the solid torus N k`1 . Furthermore, the link K k > γ k`1 is a Whitehead link for each meridian γ k`1 of N k`1 . This is a consequence of the symmetry of the Whitehead link. (3) Each K k is unknotted in S 3 . For each j ą k, K k is nullhomotopic in N j but a non-trivial knot in N j .

Remark that the Whitehead manifold has no complete metric of positive scalar curvature (See Theorem B 1 ).

1.3.3. Geometric Index. Definition 1.3.8. [Sch53] If N 1 Ă Int N are solid tori, the geometric index of N 1 in N , IpN 1 , N q, is
the minimal number of points of the intersection of the core of N 1 with a meridian disc of N .

Remark 1.3.9. If the geometric index IpN 1 , N q is greater than zero, the solid torus N is an effective handlebody relative to N 1 (See Definition 1.2.3).

If the core K 1 of N 1 is a trivial knot in N , there is a meridian disc pD, BDq Ă pN zN 1 , BN q. Moreover, IpN 1 , N q is equal to zero. (See Corollary 2.9 of [START_REF] Wang | Contractible 3-manifolds and positive scalar curvatures (I)[END_REF]) For example, in W h, the geometric index IpN k , N k`1 q " 2 for each k, where N k is illustrated as in Chapter 1.3.2.

See [START_REF] Schubert | Knoten und vollringe[END_REF] for the following results about the geometric index.

EXAMPLES

1) Let N 0 , N 1 , and N 2 be solid tori so that N 0 Ă Int N 1 and N 1 Ă Int N 2 . Then IpN 0 , N 2 q " IpN 0 , N 1 qIpN 1 , N 2 q.

2) If N 0 and N 1 are unknotted solid tori in S 3 with N 0 Ă Int N 1 , and if N 0 is homotopically trivial in N 1 , then IpN 0 , N 1 q is even.

Lemma 1.3.10. Suppose that the closed solid torus N 1 Ă Int N is homotopically trivial in the closed solid torus N . If IpN 1 , N q ą 0, then the two induced maps i 1 : π 1 pBN q Ñ π 1 pN zN 1 q and i 2 : π 1 pBN 1 q Ñ π 1 pN zN 1 q are both injective.

Proof. Since IpN 1 , N q ą 0, the solid torus N is an effective handlebody relative to N 1 (See Remark 1.3.9). We use Lemma 1.2.4 to see that the map i 1 : π 1 pBN q Ñ π 1 pN zN 1 q is injective.

Suppose that the map i 2 is not injective. We use Lemma 1.1.2 to find an embedded disc pD 2 , BD 2 q Ă pN zN 1 , BN 1 q. The embedded circle θ :" BD 2 is not contractible in BN 1 .

Since θ bounds an embedded disc D 2 Ă N zN 1 , it is a trivial knot in N . Furthermore, rθs belongs to the kernel of the map H 1 pBN 1 q Ñ H 1 pN zN 1 q. From Lemma 1.3.6, the embedded circle θ is a longitude of N 1 .

Recall that as a knot, any longitude of N 1 is isotopic to the core K 1 of N 1 in N 1 (See Remark 1.3.3). Therefore, K 1 is isotopic to θ and a trivial knot in N . From Remark 1.3.9, IpN 1 , N q " 0. This is a contradiction. Definition 1.3.12. (Genus one 3-manifold) A genus one 3-manifold M is the ascending union of solid tori tN k u kPN , so that for each k, N k Ă Int N k`1 and the geometric index of N k in N k`1 is not equal to zero.

Theorem 1.3.13. (See [Theorem 2.8, Page 2042] of [START_REF] Garity | Contractible 3-manifolds and the double 3-space property[END_REF])

(1) A genus one 3-manifold defined with a sequence of open solid tori tN k u kPN so that each N k is contractible in N k`1 , is a contractible 3-manifold that is not homeomorphic to R 3 . (2) Any contractible genus one 3-manifold can be written as an ascending union of solid tori tN k u kPN so that 1)

N k is contractible in N k`1 ; 2) IpN k , N k`1 q ě 2 for each k.
For example, the Whitehead manifold is a contractible genus one 3-manifold.

Any contractible genus one 3-manifold M :" Y 8 i"0 N k satisfies the following: (1) For each k, N k is homotopically trivial in N k`1 . Moreover, IpN k , N k`1 q ě 2.

(2) For each j ą k, the core K k of N k is null-homotopic in N j but a nontrivial knot in N j .

(3) If N j is viewed as an unknotted solid torus in S 3 , then the link K k > γ j Ă S 3 is linked in S 3 , for each meridian γ j of N j for j ą k. Moreover, its linking number is zero. (4) However, the knot K k Ă S 3 may be knotted in S 3 . Together these properties, we will show that no contractible genus one 3-manifold admits a complete metric of positive scalar curvature (See Theorem B 2 ).

More examples.

In this part, we construct two different contractible 3manifolds, M 1 and M 2 . Their fundamental group at infinity are both non-trivial.

First, we construct the example, M 1 . Choose an unknotted handlebody W 0 Ă S 3 of genus two (See Definition 1.1.14). Take a second handlebody W 1 Ă Int W 0 of genus two which is a tubular neighborhood of the curve in Figure 1.5. Then, embed another handlebody W 2 of genus two inside W 1 in the same way as W 1 lies in W 0 and so on infinitely many times. Therefore, we obtain a decreasing family tW k u k of handlebodies of genus two.

W k W k`1 Figure 1.5. The manifold M 1 is defined as M 1 :" S 3 z X 8 k"0 W k .
It is an open 3-manifold. We have that each W k is unknotted in S 3 (See Definition 1.1.14). That is, the complement N k of W k in S 3 is a handlebody of genus two. Therefore, M 1 can be written as the increasing union of handlebodies tN k u k of genus two. In addition, each N k lies in N k`1 as in Figure 1

.6. (The set K k is the core of N k .) Since each N k is homotopically trivial in N k`1 , M 1 is a contractible 3-manifold. γ k N k`1 K k γ k`1 Figure 1.6.
Next, we construct a properly embedded plane P 1 Ă M 1 and show that π 8

1 pM 1 q is non-trivial.

Choose the splitting meridian

γ k`1 Ă BN k`1 of N k`1 as in Figure 1.6. We have that γ k is homotopic to γ k`1 in N k`1 zN k . Choose an embedded annulus A k Ă N k`1 zN k (namely, it is homeomorphic to S 1 ˆr0, 1s) with boundary γ k > γ k`1 .
We define the plane P 1 as

P 1 :" Y kě0 A k Y γ 0 D 0
where D 0 Ă N 0 is a meridian disc with boundary γ 0 .

Proposition 1.3.14. Let M 1 and P 1 be constructed as above. Then, ' the fundamental group at infinity π 8 1 pM 1 q is non-trivial; ' the properly embedded plane P 1 cuts M 1 into two Whitehead manifolds.

Remark. Since π 8

1 pM 1 q is non-trivial, M 1 is not simply-connected at infinity. That is, M 1 is not homeomorphic to R 3 .

The family tN k u of handlebodies satisfies Property H (See Definition 2.2.5).

Proof. First, we will show that π 8

1 pM 1 q is non-trivial. Since γ k is homotopic to γ k`1 in M 1 zN 0 , it is sufficient to show that γ 0 is not contractible in M 1 zN 0 (See Remark 1.1.10).
We see from Figure 1.6 that N k`1 is an effective handlebody relative to N k . From Lemma 1.2.4, the map π 1 pBN k`1 q Ñ π 1 pN k`1 zN k q is injective for k ě 0.

From Figure 1.5, we have that W k is an effective handlebody relative to W k`1 . By Lemma 1.2.4, the maps π 1 pBW k q Ñ π 1 pW k zW k`1 q is also injective. In addition, the space W k zW k`1 is equal to N k`1 zN k . Then, we can conclude that the map π 1 pBN k q Ñ π 1 pN k`1 zN k q is injective for k ě 0.

Claim: the map π 1 pBN 0 q Ñ π 1 pM 1 zN 0 q is injective.

Theorem 1.1.4 gives an isomorphism between π 1 pN 2 zN 0 q and π 1 pN 1 zN 0 q˚π 1 pBN 1 q π 1 pN 1 zN 0 q. From the above fact, the maps π 1 pBN 1 q Ñ π 1 pN 1 zN 0 q and π 1 pBN 1 q Ñ π 1 pN 2 zN 1 q are both injective. We use Lemma 1.1.5 to show that the map π 1 pN 1 zN 0 q Ñ π 1 pN 2 zN 0 q is injective.

In addition, the map π 1 pBN 0 q Ñ π 1 pN 1 zN 0 q is injective. The composition of these two maps π 1 pBN 0 q Ñ π 1 pN 2 zN 0 q is injective.

Repeating the above argument several times, we obtain that the map π 1 pBN 0 q Ñ π 1 pN j zN 0 q is injective for each j ą 0. Hence, the map π 1 pBN 0 q Ñ π 1 pM 1 zN 0 q is injective. This finishes the proof of this claim. Since γ 0 is not homotopically trivial in BN 0 , it is also non-contractible in M 1 zN 0 . Since each γ k is homotopic to γ k`1 in M 1 zN 0 , each γ k is a non-trivial element in π 1 pM 1 zN 0 q. We see from Remark 1.1.10 that π 8

1 pM 1 q is non-trivial. It remains to show that P 1 cuts M 1 into two Whitehead manifolds. The plane P 1 cuts M 1 into two contractible 3-manifolds M 1 1 and M 2 1 . In addition, P 1 X N k is a splitting meridian disc of N k with boundary γ k .

From the sequence tN k u, we obtain two increasing families,

tN 1 k u and tN 2 k u, of solid tori in M 1 satisfying that ' M 1 1 " Y k N 1 k and M 2 1 " Y k N 2 k ; ' the set N k zpN 1 k > N 2 k q is a tubular neighborhood of the meridian disc P 1 X N k . Furthermore, each N 1 k is embedded into N 1 k`1 as in Figure 1.7. From Chapter 1.3.2, we see that M 1
1 is homeomorphic to the Whitehead manifold. Similarly, the contractible 3-manifold M 2 1 is also homeomorphic to the Whitehead manifold. Therefore, P 1 cuts M 1 into two Whitehead manifolds.

N 1 k N 1 k`1 Figure 1.7.
Together with the proof of Theorem B 2 , we have that Theorem 1.3.15. The contractible 3-manifold, M 1 , has no complete metric of positive scalar curvature.

We will prove it in Chapter 6.

In the following, we construct the second example, M 2 . The construction of M 2 is similar to M 1 's.

Choose an unknotted handlebody W 0 Ă S 3 of genus two. Take a second handlebody W 1 Ă Int W 0 of genus two which is a tubular neighborhood of the curve in Figure 1.8. Then, embed another handlebody W 2 of genus two inside W 1 in the same way as W 1 lies in W 0 and so on infinitely many times. Therefore, we obtain a decreasing family tW k u of handlebodies.

W k W k`1 Figure 1.8. The manifold M 2 is defined as M 2 :" S 3 z X 8 k W k . Note that each W k is unknotted in S 3
As above, M 2 can be written as an increasing union of handlebodies tN k u k of genus two, where N k :" S 3 zW k . Furthermore, each N k lies in N k`1 as in Figure 1.9.

N k`1 N k γ k`1 γ k Figure 1.9. Pick the splitting meridian γ k Ă N k of N k . Each γ k is homotopic to γ k`1 in N k`1 zN k .
Choose an embedded annulus A k Ă N k`1 zN k with boundary γ k > γ k`1 . We define the plane P 2 as

P 2 :" Y kě0 A k Y γ 0 D 0
where D 0 Ă N 0 is a meridian disc with boundary γ 0 .

Proposition 1.3.16. Let M 2 and P be constructed as above. Then, (1) π 8 1 pM 2 q is non-trivial; (2) the plane P 2 cuts M 2 into two contractible 3-manifolds. Each of them is homeomorphic to R 3

Proof. As in the proof of Proposition 1.3.14, we know that π 1 pM 8 2 q is non-trivial. It remains to show that P 2 cuts M 2 into two R 3 s. The plane P 2 cuts M 2 into two contractible 3-manifolds M 1 2 and M 2 2 . In addition, P 2 X N k is also a meridian disc of N k with boundary γ k .

As the first example, we obtain two increasing families,

tN 1 k u and tN 2 k u, of solid tori in M 2 satisfying that ' M 1 2 " Y k N 1 k and M 2 2 " Y k N 2 k ; ' the set N k zpN 1 k > N 2 k q is a tubular neighborhood of the meridian disc P 2 X N k . Furthermore, each N 1 k is embedded into N 1 k`1 as in Figure 1.10. There is a closed 3-ball B k Ă N 1 k`1 containing N 1 k . Therefore, M 1 2 is equal to Y k B k . Hence, M 1 1 is simply- connected at infinity (See Remark 1.1.8). Namely, it is homeomorphic to R 3 . Similarly, M 2
2 is also homeomorphic to R 3 .

N 1 k N 1 k`1 Figure 1.10.
Remark 1.3.17. It is unknown whether M 2 admits a complete metric of positive scalar curvature.

CHAPTER 2

Topological Properties

In this chapter, we discuss several topological properties of contractible 3-manifolds. We first study the behavior of embedded discs in the Whitehead manifold and their relationship with the geometric indexes. Their relation is clarified by Theorem 2.1.2. Based on their relation, we introduce the topological property, called Property P . Furthermore, we show that any contractible genus one 3-manifold satisfies this property.

Next, we consider contractible 3-manifolds. We introduce two types of surgeries on handlebodies. We use these surgeries to show the existence of effective handlebodies (See Theorem 2.2.3). Then we inductively find an increasing family of handlebodies with good properties, called Property (H).

Property P

2.1.1. The Whitehead case. As in Chapter 1.3.2, Wh Ă S 3 is an increasing union of closed solid tori tN k u 8 k"0 so that the geometric index IpN k , N k`1 q " 2, for each k. For any j ą k, the core K k of N k is a non-trivial knot in N j but unknotted in S 3 . In addition, the link K k > γ j is linked with zero linking number, for any meridian γ j of N j .

Lemma 2.1.1. Any embedded circle γ Ă BN k which is the boundary of a closed embedded disc D in Wh but not nullhomotopic in BN k , is a meridian of N k .

Proof. Since the disc D is compact, there is some

k 0 ą k such that D is contained in N k 0 .
Let γ belong to the homology class prγ k s `qrθ k s in H 1 pBN k q, where γ k and θ k are a meridian and a longitude of N k . Since N k is an unknotted solid torus in S 3 (See Remark 1.3.7), γ (as a knot in S 3 ) is isotopic to the torus knot T p,q in S 3 .

Because the knot γ bounds an embedded disc D in N k 0 , it is a trivial knot in N k 0 Ă S 3 . Hence, γ is unknotted in S 3 . We see from Remark 1.3.5 that p " ˘1 or q " ˘1.

Since the knot γ is trivial in N k 0 , we use Remark 1.3.9 to find a meridian disc pD 1 , BD 1 q Ă pN k 0 , BN k 0 q with D 1 X γ " H . Because the geometric index IpN k , N k 0 q ą 0, the disc D 1 contains at least one meridian γ 1 k of BN k (Lemma 1.3.11 or Lemma 1.2.5). Therefore, γ 1 k X γ is empty. Their intersection number on BN k must be zero. We knows that the intersection number of γ and γ 1 k is q. Therefore, we knows that p " ˘1, q " 0. That is to say, γ is homotopic to the meridian γ 1 k on BN k . This completes the proof.

Theorem 2.1.2. Any γ Ă BN k bounding an embedded disc D in Wh satisfies one of the following:

(1) rγs is trivial in π 1 pBN k q, (2) D X Int N l has at least IpN l , N k q components intersecting N 0 , for each l ă k.

Note that the geometric index IpN l , N k q is equal to 2 k´l .

Proof. We argue by induction on k.

' When k " 0, it is trivial.

' We suppose that it holds for N k´1 .

We suppose that the closed curve γ is not contractible in BN k . From Lemma 2.1.1, it is a meridian of N k . In addition, the linking number of γ > K k´1 is zero, where K k´1 is the core of N k´1 (See Remark 1.3.7).

We may assume that D intersects BN k´1 transversally. The set D X BN k´1 has finitely many components C :" tγ i u iPI . Each component γ i is an embedded circle and bounds a unique closed disc D i Ă Int D.

Let tγ j u jPI 0 be the set of maximal circles in C where I 0 Ă I. Each γ j is the boundary of the disc D j , for j P I 0 . Claim: There exist at least two elements in tγ j u jPI 0 , which are meridians of N k´1 .

By Lemma 1.3.10, the maps π 1 pBN k q Ñ π 1 pN k zN 0 q and π 1 pBN k q Ñ π 1 pN m zN k q are both injective for any m ą k. Van-Kampen's Theorem (See Theorem 1.1.4) gives an isomorphism between π 1 pN m zN 0 q and π 1 pN k zN 0 q ˚π1 pBN k q π 1 pN m zN k q. We use Lemma 1.1.5 to see that the map π 1 pBN k q Ñ π 1 pN m zN 0 q is also injective. Therefore, since γ is not contractible in BN k , we can conclude that it is not contractible in W hzN 0 .

If γ j is homotopically trivial in BN k´1 for each j P I 0 , then one finds a disc D 1 j Ă BN k´1 . Consider a new disc D 1 :" pΣz Y jPI 0 D j q Y pY jPI 0 D 1 j q in W hzN 0 with boundary γ. Therefore, γ is contractible in W hzN 0 . This contradicts the last paragraph. We see that one of tγ j u jPI 0 is non-contractible in BN k´1 . Hence, by Lemma 2.1.1, there is at least one meridian of N k´1 in tγ j u jPI 0 .

In the following, we argue by contradiction. Suppose that there is a unique meridian of N k´1 in the set tγ j u jPI 0 . That is to say, there is a unique j 0 P I 0 such that γ j 0 is a meridian of N k´1 . Remark that each γ j bounds a unique disc D j Ă D.

If γ j is not contractible BN k for some j P I 0 ztj 0 u, Lemma 2.1.1 shows that it is a meridian, which contradicts the uniqueness of j 0 . We see that γ j is nullhomotopic in BN k´1 , for each j P I 0 ztj 0 u.

Consider a meridian disc Dj 0 of N k´1 with boundary γ j 0 , which intersects the core K k´1 of N k´1 transversally at one point. For j P I 0 ztj 0 u, there exists a disc Dj Ă BN k´1 with boundary γ j .

Define a new disc D :" pDz Y jPI 0 D j q Y jPI 0 pY γ j Dj q with boundary γ. It intersects K k´1 transversally at one point, which implies that the intersection number of D and K k´1 is ˘1.

Therefore, the linking number of γ > K k´1 is ˘1. This is in contradiction with the fact that its linking number is zero.

This completes the proof of the claim.

From the above claim, there are at least two distinct meridians, γ j 0 and γ j 1 , of N k´1 in tγ j u jPI 0 . Applying our inductive assumption to D j 0 and D j 1 respectively, we know that D jt X Int N l has at least 2 k´1´l components intersecting N 0 for t " 0, 1 for l ď k ´1. Therefore, D X Int N l has at least 2 k´l components intersecting N 0 .

Based on Theorem 2.1.2, we introduce a topological property.

Definition 2.1.3. A contractible genus one 3-manifold M is called to satisfy Property P if for any properly embedded plane Σ Ă M , any k ą 0 and any closed curve γ Ă BN k XΣ, it holds one of the following:

(1) γ is contractible in BN k ;

(2) for l ă k, D X Int N l has at least IpN l , N k q components intersecting N 0 , where D Ă Σ is a unique disc with boundary γ and tN k u k is a sequence as described in Theorem 1.3.13.

We will show that all contractible genus one 3-manifolds satisfy Property P (Theorem 2.1.6).

2.1.2. The Genus one case. In this part, we show that any contractible genus one 3-manifold satisfies Property P .

First, recall some notations from Chapter 1.3.4. Any contractible genus one 3-manifold M is the ascending union of closed solid tori tN k u 8 k"0 so that N k is homotopically trivial in N k`1 and the geometric index IpN k , N k`1 q ě 2 (See Theorem 1.3.13).

In the genus one case, Lemma 2.1.1 can be generalized as follows:

Lemma 2.1.4. A circle γ Ă Σ X BN k , which is not contractible in BN k , is a meridian of N k ,
where Σ Ă M is a properly embedded plane. Moreover, the unique disc D Ă Σ with boundary γ intersects the core K 0 of N 0 .

Proof. We may assume that Σ intersects BN k transversally. Since Σ is properly embedded, Σ X BN k :" tγ i u n i"0 has finitely many components, where γ 0 " γ. Each γ i bounds a unique closed disc D i Ă Σ (where D 0 " D).

Define the set C :" tγ i |γ i Ă D 0 is not contractible in BN k u. It is not empty (γ 0 Ă D 0 ). Since tγ i u n i"0 is a family of disjoint circles, we see that the intersection number of γ and γ i in BN k is zero for each i ‰ 0.

If rγ i s is not equal to ˘rγs in π 1 pBN k q for some γ i P C, the intersection number of γ and γ i is nonzero in BN k . This contradicts the above fact. We can conclude that each γ i P C is homotopic to γ in BN k , up to orientation.

In the following, we will show that each minimal circle γ j in C is a meridian. This is to say, γ is also a meridian of N k .

The remaining proof is similar to the proof of Lemma 1.2.4. It is sufficient to show that γ j is homotopically trivial in N j . We begin by constructing an immersed disc Dj Ă N k with boundary γ j .

Let us consider the set C j :" tγ i |γ i Ă Int D j u Ă C and the set C max j of maximal circles in C j . One has two cases: C j " H or C j ‰ H.

Case I: If C j is empty, we consider the set Z :" Int D j and define the disc Dj as Int D j ;

Case II: If C j is not empty, then C max j is also non-empty. From the minimality of γ j , each

γ i P C max j is contractible in BN k and bounds a disc D 1 i Ă BN k . Define the set Z :" Int D j z Y γ i PC max j D i and the disc Dj :" Z Y pY γ i PC max j D 1 i q with boundary γ j .
Let us explain why Dj is contained in N k . In any case, BN k cuts M into two connected components, M zN k and Int N k . The set Z is one of these components of Int D j zBN k .Therefore, it is in M zN k or Int N k .

If Z is in M zN k , then the disc Dj with boundary γ j is contained in M zN k . Therefore, we see that rγ j s " 0 in π 1 pM zN k q. However, the map π 1 pBN k q Ñ π 1 pM zN k q is injective (Lemma 1.3.10). That is to say, γ j is null-homotopic in BN k . This contradicts the fact that rγ j s ‰ 0 in π 1 pBN k q. We can conclude that Z is contained in Int N k .

Therefore, Dj is contained in

N k . Its boundary γ j is nullhomotopic in N k . Since γ is homotopic to γ j in BN k , it is also contractible in N k . By Definition 1.2.1, γ must be a meridian of N k .
By Lemma 1.3.10, the two induced maps π 1 pBN k q Ñ π 1 pM zN k q and π 1 pBN k q Ñ π 1 pN k zK 0 q are both injective. Van-Kampen's theorem (See Theorem 1.1.4) shows that π 1 pM zK 0 q -π 1 pM zN k q ˚π1 pBN k q π 1 pN k zK 0 q. We see from Lemma 1.1.5 that the map π 1 pBN k q Ñ π 1 pM zK 0 q is also injective. Therefore, rγs ‰ 0 in π 1 pM zK 0 q. We can conclude that the disc D Ă Σ with boundary γ must intersect the core K 0 of N 0 .

Remark 2.1.5. ' In the proof, the set Dj X Int N k is equal to the set Z and a subset of D X Int N k . ' The disc Dj may not be embedded, because D 1 i may be contained in some D 1 i 1 . When it is not an embedding, we can deform Dj in a small neighborhood of BN k in N k so that it becomes an embedded disc in N k .

Theorem 2.1.6. Any contractible genus one 3-manifold M satisfies Property P .

Proof. Consider a properly embedded plane Σ Ă M . Suppose there is some closed curve γ Ă Σ X BN k which is not contractible in BN k for some k P N ą0 . By Lemma 2.1.4, γ is a meridian of N k and the unique closed disc D Ă Σ with boundary γ intersects N 0 .

We may assume that Σ intersects BN k transversally. The set D X BN k :" tγ i u n i"0 has finitely many components where γ 0 " γ.

Define the set C :"

tγ i | the circle γ i Ă D X BN k is not contractible in BN k u. (It is not empty because γ Ă D). We use Lemma 2.1.4 to see that each minimal circle γ j in C is a meridian of N k . It bounds a unique closed disc D j Ă D.
As in the proof of Lemma 2.1.4, we construct a disc Dj Ă N k with boundary γ j . Remark that Dj X IntN k is a subset of D X IntN k (See the above Remark).

As described in the above remark, the disc Dj may be not embedded. If necessary, we can deform it in a small neighborhood of BN k in N k so that it becomes an embedded disc. For l ă k, Dj X Int N l is still a subset of D X Int N l Ă Σ.

It is sufficient to show that Dj X Int N l has at least IpN l , N k q components intersecting N 0 .

We may assume that Dj intersects BN l transversally. The intersection Dj X BN l :" tγ 1 t u tPT has finitely many components. Let us consider the set Ĉmax of maximal circles in tγ 1 t u tPT and its subset Ĉnon :"

tγ 1 t P Ĉmax |γ 1 t is not contractible in BN l u. Claim: | Ĉnon | ě IpN l , N k q.
We argue by contradiction. Suppose that

| Ĉnon | ă IpN i , N k q. Each γ 1 t P Ĉmax bounds a unique disc D 1 t Ă Dj . If γ 1
t is in Ĉnon , it is a meridian of N l (See Lemma 2.1.4). Therefore, we can find a meridian disc D 2 t of N l which intersects the core K l of N l transversally at one point. If

γ 1 t P Ĉmax z Ĉnon , γ 1 t is contractible in BN l and bounds a disc D 2 t in BN l . Define a disc D1 j with boundary γ j D1 j :" p Dj z Y γ 1 t P Ĉmax D 1 t q Y pY γ 1 t P Ĉmax D 2 t q.
The number #p D1 j X K l q of points of D1 j X K l is less than IpN l , N j q. As above, the disc D1 j may be not embedded (because D 2 t may be contained in some D 2 t 1 ). If necessary, we modify the disc D1 j in a small neighborhood of BN l so that it becomes an embedded disc in N k .

Therefore, we may assume that p D1 j , B D1 j q Ă pN k , BN k q is an embedded disc with boundary γ j . Since γ j is a meridian of N k (See Lemma 2.1.4), D1 j is a meridian disc of N k with #p D1 j X K l q ă IpN l , N k q. However, the definition of the geometric index (See Definition 1.3.8) gives that #p D1 j X K l q ě IpN l , N k q, a contradiction. This finishes the proof of the claim.

In the following, we will finish the proof of the theorem. Let tγ 1 s u m s"1 be the circles in Ĉnon and D 1 s Ă Dj the unique disc with boundary γ 1 s , where m " | Ĉnon j |. From the maximality of γ 1 s in tγ 1 t u tPT , tD 1 s u m s"1 is a family of pairwise disjoint discs in Dj .

We use Lemma 2.1.4 to see that each γ 1 s P Ĉnon is a meridian. Thus, D 1 s intersects the core K 0 of N 0 . The intersection D 1 s X Int N l contains at least one component intersecting N 0 .

We conclude that Dj X Int N l has at least m components intersecting N 0 . From the above claim, we know that m ě IpN l , N k q. Therefore, D X Int N l has at least IpN l , N k q components intersecting N 0 .

Remark 2.1.7.

' The proof of Lemma 2.1.4 and Theorem 2.1.6 just depend on the injectivity of the two maps π 1 pBN k q Ñ π 1 pN k zN 0 q and π 1 pBN k q Ñ π 1 pM zN k q. ' Let tR k u k be an increasing family of solid tori in a contractible 3-manifold with the property that (1) π 1 pBR k q Ñ π 1 pR k zR 0 q is injective ;

(2) π 1 pBR k q Ñ π 1 pM zR k q is injective . However, the union Y k R k may not be equal to M . From the above fact, Lemma 2.1.4 holds for each R k . Further, as in the proof of Theorem 2.1.6, the family tR k u satisfies Property P. That is to say, for any properly embedded plane Σ Ă M , any k ą 0 and any closed curve γ Ă BR k X Σ, it holds one of the following:

(1) γ is contractible in BR k ;

(2) for l ă k, D X Int R l has at least IpR l , R k q components intersecting R 0 , where D Ă Σ is a unique disc with boundary γ. Type II: If there exists an embedded disc D 1 Ă M zN satisfying that 1) Int D 1 Ă M zN and 2) its boundary γ Ă BN is not contractible in BN , we consider a closed tubular neighborhood

Property

N 1 pD 1 q of D 1 in M zN . Define a new handlebody W 2 as N Y N 1 pD 1 q.
Remark 2.2.1. For i " 1, 2, the genus gpBW i q of BW i is less than gpBN q. In addition, BW i is a union of BW i X BN and some disjoint discs. It tells us that the map π 1 pBW i X BN q Ñ π 1 pBW i q is surjective.

Lemma 2.2.2. If N 1 is homotopically trivial in N , then N 1 is also homotopically trivial in W i for each i, where W i is obtained from the above surgeries.

Proof. For the type II surgery, we see that N is contained in W 2 . Therefore, N 1 is homotopically trivial in W 2 .

For the type I surgery, it is sufficient to show that any circle c Ă N 1 bounds some disc D1 Ă W 1 .

The closed curve c bounds an immersed disc D 1 Ă Int N . We will construct the required disc D1 Ă W 1 from D 1 .

We may assume that

D 1 intersects D ´> D `:" Int N X BN pD 1 q transversally. Each component c i of D 1 X pD `> D ´q is a circle in D 1 and bounds a closed sub-disc D 1 i Ă D 1 . Since D `and D ´are two disjoint discs, each c i is contractible in D `> D ´. It also bounds a disc D 2 i Ă D `> D ´. Let C max be the set of the maximal circles of tc i u iPI in D 1 . We construct a disc D1 :" D 1 z Y c i PC max D 1 i Y pY c i PC max D 2 i q with boundary c. It stays in N zN pD 1 q. That is to say, c is contractible in W 1 . Therefore, N 1 is homotopically trivial in W 1 .
2.2.2. Effective Handlebodies. In the following, let us consider a contractible 3manifold M .

Theorem 2.2.3. Let N 1 and N be two closed handlebodies in M satisfying that 1) N 1 Ă Int N and 2) N 1 is homotopically trivial in N .Then there exists a closed handlebody R Ă M containing N 1 satisfying that (1) the map π 1 pBRq Ñ π 1 pRzN 1 q is injective;

(2) the map π 1 pBRq Ñ π 1 pM zRq is injective;

(3) N 1 is homotopically trivial in R;

(4) BR is a union of BR X BN and some disjoint discs.

Remark. From (1), R is an effective handlebody relative to N 1 (Lemma 1.2.4).

Proof. Suppose that either the map i 1 : π 1 pBN q Ñ π 1 pN zN 1 q is not injective or the map i 2 : π 1 pBN q Ñ π 1 pM zN q is not injective. (If these two maps are both injective, R is defined as N .)

If i 1 is not injective, Lemma 1.1.2 shows that there exists a meridian disc D 1 of N with D 1 X N 1 " H. We do the type I surgery on N with the disc D 1 to obtain a new handlebody W .

If i 2 is not injective, we use Lemma 1.1.2 to find an embedded circle γ Ă BN and an embedded disc D 2 Ă M zN (Int D 2 Ă M zN ) where γ " BD 2 is not nullhomotopic in BN . We do the type II surgery with the disc D 2 to get a new handlebody W .

In any case, we have that gpBW q ă gpBN q. The boundary BW is a union of BW X BN and some disjoint discs tD 1 i u i . Therefore, π 1 pBW X BN q Ñ π 1 pBW q is surjective. In addition, we see from Lemma 2.2.2 that N 1 is contractible in W .

When picking a circle γ Ă BW which is not nullhomotopic in BW , we may assume that γ is an embedded circle in BW X BN . Therefore, when repeating these two types of surgeries, we may assume that the new surgeries are operated away from these disjoint discs tD 1 i u. Iterate this process until we find a handlebody R satisfying (1) and (2). At each step, the genus of the handlebody obtained from the surgery is less than the original one. Therefore, this process stops in no more than gpN q steps. As above, N 1 is contractible in R and BR is a union of BR X BN and some disjoint discs.

Remark. If N 1 is not contained in a 3-ball in M , then the genus of R is greater than zero.

Lemma 2.2.4. Let R Ă M be a closed effective handlebody relative to the closed handlebody N 1 Ă Int R satisfying that π 1 pBRq Ñ π 1 pM zRq is injective. If a closed handlebody N is an effective handlebody relative to R Ă Int N , then N is an effective handlebody relative to N 1 .

Proof. Based on Lemma 1.2.4, it is sufficient to show that the map π 1 pBN q Ñ π 1 pN zN 1 q is injective.

We use Lemma 1.2.4 to show that the induced map π 1 pBRq Ñ π 1 pRzN 1 q is injective. Since π 1 pBRq Ñ π 1 pM zRq is injective, then the map π 1 pBRq Ñ π 1 pN zRq is also injective.

Van Kampen's theorem (Theorem 1.1.4) gives an isomorphism between π 1 pN zN 1 q and π 1 pN zRq ˚π1 pBRq π 1 pRzN 1 q. A classical result (See [Theorem 11.67, Page 404] of [START_REF] Joseph J Rotman | An introduction to the theory of groups[END_REF] or Lemma 1.1.5) shows that the induced map π 1 pN zRq Ñ π 1 pN zN 1 q is injective. Lemma 1.2.4 shows that the map π 1 pBN q Ñ π 1 pN zRq is injective. Therefore, the composition π 1 pBN q Ñ π 1 pN zRq Ñ π 1 pN zN 1 q is also injective. This finishes the proof.

2.2.3. Property H. In the following, let us consider a contractible 3-manifold M which is not homeomorphic to R 3 .

By Theorem 1.1.12, M can be written as an ascending union of handlebodies tN k u 8 k"0 . Each N k is contractible in N k`1 . As in Remark 1.1.8, we can choose N 0 so that it is not contained in a 3-ball in M (because M is not homeomorphic to R 3 ).

In the genus one case, the family tN k u has several good properties. For example, each N k is an effective handlebody relative to N 0 and the map π 1 pBN k q Ñ π 1 pM zN k q is injective (See Lemma 1.3.10 or Lemma 2.10 of [START_REF] Wang | Contractible 3-manifolds and positive scalar curvatures (I)[END_REF]). These properties are necessary and crucial in our proof. In general, the family tN k u may not have these properties. To overcome this difficulty, we introduce a topological property, called Property H. Definition 2.2.5. A family tR k u k of handlebodies in a contractible 3-manifold M :"

Y k N k is called to have Property H if it satisfies that
(1) the map π 1 pBR k q Ñ π 1 pR k zR 0 q is injective for k ą 0;

(2) the map π 1 pBR k q Ñ π 1 pM zR k q is injective for k ě 0;

(3) each R k is contractible in R k`1 but not contained in a 3-ball in M ; (4) there exists a sequence of increasing integers tj k u k , such that π 1 pBR k X BN j k q Ñ π 1 pBR k q is surjective. where tN k u is assumed as in Remark 1.1.13. For example, in a contractible genus one 3-manifold M :" Y k N k , the family tN k u k satisfies Property H, where tN k u is assumed in Chapter 1.3.4 (See Lemma 1.3.10).

In the following, we will prove if a contractible 3-manifold M is not homeomorphic to R 3 , there is a family of handlebodies with Property H (See Theorem 2.2.6). However, such a family is not unique . Theorem 2.2.6. If a contractible 3-manifold M :" Y k N k (as above) is not homeomorphic to R 3 , then there is an ascending family tR k u k of closed handlebodies in M satisfying that

(1) the map π 1 pBR k q Ñ π 1 pR k zR 0 q is injective for k ą 0;

(2) the map π 1 pBR k q Ñ π 1 pM zR k q is injective for k ě 0;

(3) each R k is contractible in R k`1 but not contained in a 3-ball in M ; (4) there exists a sequence of increasing integers tj k u k , such that π 1 pBR k X BN j k q Ñ π 1 pBR k q is surjective.

Remark 2.2.7. ' The union Y k R k may be not equal to M . ' For k ą 0, Van-Kampen's Theorem gives an isomorphism between π 1 pM zR 0 q and π 1 pM zR k q ˚π1 pBR k q π 1 pR k zR 0 q. Based on (1) and (2) in Theorem 2.2.6, we use Lemma 1.1.5 to show that the map π 1 pBR k q Ñ π 1 pM zR 0 q is injective. ' As (4) in Theorem 2.2.3, BR k is the union of BR k X BN j k and disjoint discs.

Proof. First, we construct R 0 . We repeatedly apply the Type II surgery to N 0 , until we find a handlebody R 0 containing N 0 so that π 1 pBR 0 q Ñ π 1 pM zR 0 q is injective.

From Remark 2.2.1, we see that, at each step, the genus of the handlebody obtained from the surgery is less than the original one. Therefore, this process stops in no more than gpN 0 q steps. In addition, since N 0 is not contained in a 3-ball in M , then R 0 has the same property.

It remains to construct the sequence tR k u k inductively. When k is equal to 1, we pick a handlebody N j 1 containing R 0 satisfying that R 0 is homotopically trivial in N j 1 . Its existence is ensured by the following fact:

Because R 0 is compact, there is some handlebody

N j 1 ´1 containing R 0 . Since N j 1 ´1 is homotopically trivial in N j 1 , R 0 is contained in N j 1 and contractible in N j 1 .
By Theorem 2.2.3, there exists a handlebody R 1 containing R 0 so that

' π 1 pBR 1 q Ñ π 1 pR 1 zR 0 q is injective; ' π 1 pBR 1 q Ñ π 1 pM zR 1 q is injective; ' R 0 is contractible in R 1 ;
' BR 1 is a union of BR 1 X BN j 1 and some disjoint closed discs. Therefore, π 1 pBR 1 X BN j 1 q Ñ π 1 pBR 1 q is surjective. In particular, since R 0 is not contained in a 3-ball in M , R 1 has the same property.

Suppose that there exists a handlebody R k´1 and a positive integer j k´1 satisfying (1), (2), (3) and (4) in Theorem 2.2.6.

As the existence of N j 1 , there exists a handlebody N j k containing R k´1 satisfying that R k´1 is homotopically trivial in N j k . We use Theorem 2.2.3 to find an effective handlebody R k relative to R k´1 satisfying (2), (3) and (4).

Since the map π 1 pBR k´1 q Ñ π 1 pR k´1 zR 0 q is injective, R k´1 is an effective handlebody relative to R 0 (Lemma 1.2.4). Lemma 2.2.4 shows that R k is an effective handlebody relative to R 0 . We apply Lemma 1.2.4 again and get that R k also satisfies (1). This finishes the proof.

Part 2

Minimal Surfaces CHAPTER 3

Minimal Surfaces

A minimal surface is a submanifold in a Riemannian manifold whose mean curvature is identically zero. It is also the critical point of the area functional (See the first variation formula, Equation (3.1.12)).

In Section 3.1, we first introduce the so-called first and second variation formulas for the area functional. Subsequently, we derive some notations, such as minimal surfaces, Morse index and the stability condition. Finally, we focus on the Plateau Problem and related results.

In Section 3.2, we discuss some local properties of minimal surfaces, including the strong maximal principle (See Corollary 3.2.3) and the monotonicity formula (See Proposition 3.2.5) for the area. Particularly, the monotonicity formula gives a quantitative estimate for the area (See Corollary 3.2.6). This estimate can be generalized to the Riemannian case (See Theorem 3.2.7).

In Section 3.3, we study the topology of stable minimal surfaces. The stable minimal hypersurface is characterized by the first eigenvalue of the stable operator (See Lemma 3.3.1 and Theorem 3.3.4). In a manifold of positive scalar curvature, there are many topological constraints for stable minimals surfaces (See Proposition 3.3.5). For example, in the case of 3-manifolds with positive scalar curvature, the geometry of stable minimal surfaces is influenced by the extrinsic version of Cohn-Vossen inequality (See Corollary 3.3.6 and Theorem 3.3.10). As an application, we give a new proof of Theorem 2 of [START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF]. Finally, we use Theorem 3.3.8 to study contractible 3-manifolds whose scalar curvature has a decay at infinity. In the following, if X is a vector field on Σ Ă M , then we let X T and X N denote the tangential and normal components, respectively. The covariant derivative ∇ on M induces a covariant derivative ∇ Σ on Σ and the second fundamental form A of Σ. More precisely, the induced covariant derivative ∇ Σ is given by (3.1.1) ∇ Σ " p∇q T and the vector-valued bilinear form A on Σ is given for X, Y P T Σ by

(3.1.2) ApX, Y q " p∇ X Y q N .
Since the Lie bracket of X and Y is a tangential vector field in T Σ, it is easy to see that A is symmetric, i.e., ApX, Y q " ApY, Xq.

The mean curvature vector H at a point x P Σ is defined

H " k ÿ i"1 ApE i , E i q,
where tE i u is an orthonormal basis for T x Σ. Furthermore, the squared norm of the second fundamental form at x is given by

(3.1.3) |A| 2 " k ÿ i,j"1 |ApE i , E j q| 2 .
Recall also that the Gauss equations assert if X, Y P T x Σ, then

K Σ pX, Y q|X ^Y | 2 " K M pX, Y q|X ^Y | 2 `gpApX, Xq, ApY, Y qq ´gpApX, Y q, ApX, Y qq (3.1.4)
where |X ^Y | 2 is given by |X ^Y | 2 " gpX, XqgpY, Y q ´gpX, Y q 2 and K X pX, Y q and K Σ pX, Y q are the sectional curvatures of M and Σ , respectively, in the 2-plane spanned by X and Y .

For example, let Σ n´1 Ă M n be a hypersurface and N a unit normal vector field in a neighborhood of x P Σ, then ∇ p¨q N : T x Σ Ñ T x Σ is a symmetric map (often referred to as the Weingarten map) and its eigenvalues tκ i u n´1 i are called the principle curvatures. Moreover, gpH, N q " ´n´1 ÿ

i"1

κ i .
Finally, if X is a vector field over Σ, then the divergence of X at x P Σ is defined as

(3.1.5) div Σ X " n´1 ÿ i"1 gp∇ T E i X, E i q
where tE i u is an orthonormal basis for T x Σ. Notice that div Σ satisfies the Leibniz rule

div Σ pf Xq " gp∇ Σ f, Xq `f div Σ pXq.
We can also use div Σ to define the Laplace operator ∆ Σ on Σ by

∆ Σ f " div Σ p∇ Σ f q. A function f is a harmonic function on Σ if ∆ Σ f " 0. Remark 3.1.1. Note that div Σ Y N " ÿ i gpE i , ∇ E i Y N q " ´ÿ i gpY N , ∇ E i E i q
" ´gpY N , Hq.

(3.1.6) 3.1.2. First Variation Formula. Let F : Σ k ˆp´ , q Ñ M n be a variation of a k-dimensional submanifold Σ k with compact support and fixed boundary. That is, F " Id outside a compact set, F px, 0q " x and for all x P BΣ, F px, tq " x. The vector F t restricted to Σ is called the variational vector field. Now we want to compute the first variation of area for this one parameter family of surfaces. Let px i q be a local coordinate on Σ. Set

g ij ptq " gpF x i , F x j q, νptq " b detpg ij ptqq a detpg ij p0qq, (3.1.7)
where pa ij q denotes the inverse of the metric pa ij q and 1 ď i, j ď k. Further, the area formula is We may choose an orthonormal coordinate system, i.e. so that at the point x

g ij p0q " δ ij " $ & % 0, i ‰ j; 1, i " j.
Using that rF t , F x i s " 0, under this coordinate, we get at x, " div Σ pF t q " div Σ pF T t q `div Σ pF N t q " ´gpH, F N t q `div Σ pF T t q.

d dt det pg ij p0qqptq ˇˇˇt "0 " ÿ ij d dt g ij ptq ˇˇˇt "0 g ij p0q " k ÿ i"1 d dt pgpF x i , F x i qq ˇˇˇt "0 " 2 k ÿ i"1 gp∇ Ft F x i , F x i q " 2 k ÿ i"1 gp∇ Fx i F t , F x i q " 2div Σ pF t q. ( 3
(3.1.11)

Integrating Equation (3.1.11) gives the so-called first variation formula:

(3.1.12)

d dt VolpF pΣ, tqq " ´żΣ gpF N t , Hq " ż Σ div Σ pF t q.
Note that Stokes' Theorem was used to see that ş Σ div Σ pF T t q " 0. As a consequence of Equation (3.1.12), we see that Σ is a critical point for the area functional if and only if the mean curvature H vanishes identically. Definition 3.1.2. An immersed submanifold Σ k Ă pM n , gq is said to be minimal if the mean curvature H vanishes identically.

For example, let Σ be the graph of a function u : R n Ñ R. The hypersurface Σ is minimal in R n`1 if and only if u satisfies

(3.1.13) divp ∇u a 1 `|∇u| 2 q " 0.
It is the so-called minimal surface equation. Furthermore, in dimension three (n " 2), we know

(3.1.14) |Hesspuq| 2 p1 `|∇u| 2 q 3 ď |A| 2 ď 2 |Hesspuq| 2 1 `|∇u| 2
See Pages 28 and 29 in [START_REF] Tobias | A course in minimal surfaces[END_REF].

3.1.3. Second Variation Formula. Suppose now that Σ k Ă pM n , gq is a minimal submanifold. We want to compute the second derivative of the area functional for a variation of Σ. Therefore, we consider a variation F of Σ with compact support. In fact, we assume that F is a normal variation, that is, on Σ we have F T t p¨, 0q " 0. As before, let px i q i be a local coordinate on Σ and set

g ij ptq " gpF x i , F x j q, νptq " b detpg ij ptqq a detpg ij p0qq.
Differentiating the measure νptq gives

(3.1.15) d 2 dt 2 VolpF pΣ, tqq ˇˇˇt "0 " ż Σ d 2 dt 2 νptq ˇˇˇt "0 b detpg ij p0qq.
Recall that the first derivative of the measure νptq can be written as

(3.1.16) 2 d dt νptq " ÿ ij g 1 ij ptqg ij ptqνptq.
To evaluate d 2 dt 2 νptq ˇˇˇt "0 at some point x P Σ, we may choose an orthonormal coordinate px i q i at x. Since the metric pg ij q is the identity at x, the vectors F x i give an orthonormal basis for T Σ at x. Differentiating Equation (3.1.16) then gives at x,

(3.1.17) 2 d 2 dt 2 νptq ˇˇˇt "0 " ÿ i g 2 ii p0q ´ÿ ij g 1 ij p0q 2 `1{2p ÿ i g 1 ii p0qq 2 .
Therefore, we get at x

(3.1.19) d 2 dt 2 νptq ˇˇˇt "0 " ´|gpAp¨, ¨q, F t q| 2 `|∇ N Σ F t | 2 ´Tr Σ gpR M p¨, F t q¨, F t q `div Σ pF tt q.
Note that we used the skew symmetry of R M to reverse the sign. The vector F tt p¨, 0q can be decomposed into two parts, the tangential part F T tt and the normal partial part F N tt . We use Stokes' theorem to see that (3.1.20)

ş Σ div Σ pF T tt q " 0.
The self-adjoint operator L is the so-called stability operator (or Jacobi operator ) defined on a normal vector field X to Σ by LpXq " ∆ N Σ X `Tr Σ gpR M p¨, Xq¨, Xq `ÂpXq where  is Simons' operator defined by ÂpXq " k ÿ i,j"1 gpApE i , E j q, XqApE i , E j q and ∆ N Σ is the Laplacian on the normal bundle, that is

∆ N Σ X " k ÿ i"1 p∇ E i ∇ E i Xq N ´k ÿ i"1 p∇ p∇ E i E i q T Xq N .
A normal vector field X with LpXq " 0 is said to be a Jacobi field.

We will adopt the convention that λ is a (Dirichlet) eigenvalue of L on Ω Ă Σ if there exists a non-trivial normal vector field X which vanishes on BΩ so that LpXq `λX " 0. Definition 3.1.4. The Morse index of a compact minimal surface Σ k Ă pM n , gq is the number of negative eigenvalues of the stability operator L (counting with multiplicity) acting on the space of smooth sections of the normal bundle which vanishes on the boundary.

A minimal surface submanifold Σ k Ă pM n , gq is stable if for all variations F with fixed boundary and compact support 3.1.4. Existence of Minimal surfaces. The following fundamental existence problem for minimal surfaces is known as the Plateau problem:

Given a closed curve Γ, find a minimal surface with boundary Γ. This problem was first formulated by Lagrange in 1760 and was studied extensively by Plateau in the 19th century. This question had led to many significant developments in partial differential equations and geometric measure theory, such as Morrey's works [START_REF] Charles B Morrey | The problem of plateau on a riemannian manifold[END_REF][START_REF] Bradfield | Multiple integrals in the calculus of variations[END_REF] and Fleming-Federer's works [START_REF] Federer | Normal and integral currents[END_REF].

There are various solutions to this problem, depending on the exact definition of a surface (parametrized disc, integral current, Z 2 -current or varifold ). In the following, we consider the version of the Plateau problem for parametrized discs.

In the case of R 3 , the solution was obtained in 1930 by J.Douglas [START_REF] Douglas | Solution of the problem of plateau[END_REF] and simultaneously by T.Radó [START_REF] Radó | On plateau's problem[END_REF].

Theorem 3.1.6. (See [Theorem 4.1, Page 134] in [START_REF] Tobias | A course in minimal surfaces[END_REF]) Given a piecewise C 1 closed Jordan curve Γ Ă R 3 . there exists a map u : D Ă R 2 Ñ R 3 so that (1) u : BD Ñ Γ is monotone and onto;

(2) u P C 0 pDq X W 1,2 pDq and is C 8 in the interior of D;

(3) The image of u minimizes area among all maps from the discs with boundary Γ.

The generalisation to the Riemannian manifold is due to C.B.Morrey [START_REF] Charles B Morrey | The problem of plateau on a riemannian manifold[END_REF][START_REF] Bradfield | Multiple integrals in the calculus of variations[END_REF].

For fixed boundary Γ, an area-minimizing disc is a solution to the Plateau Problem. Therefore, Morrey [Mor48, Mor09] used the variation method to find such a disc in a homegenous Riemannian 3-manifold. That is, take a sequence of mappings from the disc to the 3-manifold whose area are going to the infimum and attempt to extract a convergent subsequence. Morrey [Mor48, Mor09] pointed out that the limit of a subsequence of mappings is also the solution to the Dirichlet problem for the harmonic map.

For example, in R 3 , the existence of weak (W 1,2 -)solution to the Dirichlet problem for the harmonic map is ensured by the Kondrachov compactness theorem for W 1,2 (See [Theorem 7.22, Page 167] of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]). Weyl's Lemma told us that the weak solution is smooth in the interior of domain. This result also follows from the standard regularity theory (See [Theorem 2.10, Page 23] of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]). The above argument can be generalized to a homogenous Riemannian 3-manifold. This solution is called Morrey's solution to the Plateau problem. In addition, the argument of Gulliver [START_REF] Robert | Regularity of minimizing surfaces of prescribed mean curvature[END_REF] and Osserman [START_REF] Osserman | A proof of the regularity everywhere of the classical solution to plateau's problem[END_REF] pointed out that this solution has no interior branched point.

The remaining issue is the regularity of this solution up to the boundary. In the case of R 3 , J.J. Nitsche [START_REF] Nitsche | The boundary behavior of minimal surfaces. kellogg's theorem and branch points on the boundary[END_REF] gave an answer:

ă α ă 1, then u is C k,α on D.
This result can be generalized to the Riemannian 3-manifolds in [START_REF] Heinz | Some remarks on minimal surfaces in riemannian manifolds[END_REF].

In various works, compact 3-manifolds with mean convex boundary were intensively studied. Let pM, gq be a compact 3-manifold with boundary. The boundary BM is mean convex if ' BM is a piecewise smooth 2-manifold consisting of smooth surfaces tH i u i ' for each i, the mean curvature of H i is nonnegative.

Theorem 3.1.8. (See [START_REF] Meeks | Topology of three dimensional manifolds and the embedding problems in minimal surface theory[END_REF][START_REF] Meeks | The existence of embedded minimal surfaces and the problem of uniqueness[END_REF] or[Theorem 6.28, Page 224] of [START_REF] Tobias | A course in minimal surfaces[END_REF] ) Let pM 3 , gq be a compact Riemannian 3-manifold whose boundary is mean convex and γ a simple closed curve in BM which is null-homotopic in M . Then, γ bounds an areaminimizing disc and any such least area disc is properly embedded. This theorem will be repeatedly used in the following.

Local structures of Minimal Surfaces

It is classical that the minimal surface theory consists of two aspects: PDE and Geometry. These two aspects give various results about local structures of minimal surfaces, such as the maximum principle and the monotonicity formula.

3.2.1. Minimal surfaces are locally graphical. We begin with a minimal surface Σ Ă R 3 . The Gauss map is a continuous choice of a unit normal

N : Σ Ñ S 2 Ă R 3 .
There are two choices of such a map N and ´N corresponding to a choice of orientation of Σ. Suppose that E 1 ,E 2 is an orthonormal frame on Σ. We know that

ă ∇ E i N, E j ą" ´AΣ pE i , E j q
where A Σ is the second fundamental form (See Equation ( 3

d x,y ď 1 2 ă π 4 .
Therefore, it follows that B Σ px, 2sq is contained in the graph of a function u over a subset of T x Σ.

We may choose a coordinate px k q 3 k"1 on R 3 so that x " p0, 0, 0q,

T x Σ " tpx 1 , x 2 , 0qu.
Therefore, N pxq " p0, 0, 1q and N pyq "

p´ux 1 ,´ux 2 ,1q
?

1`|∇u| 2 . From Equation (3.2.3), we have p1 `|∇u| 2 q "ă N x , N y ą ´2" cos ´2pd x,y q where the last equality comes from Equation (3.2.1).

If y P B Σ px, 2sq, Equation (3.2.5) implies that |∇upyq| ď 1. The Hessian estimate of u comes from the gradient and curvature estimate together with Equation (3.1.14):

|Hesspuq| 2 ď p1 `|∇u| 2 q 3 |A| 2 ď 1 2 s ´2.
3.2.2. Strong Maximal Principle. First note that the difference of two solutions to the minimal surface equation (See Equation (3.1.13)) satisfies an elliptic divergent form equation (where the bound on the ellipticity depends on the gradient of the minimal graphs). Proof. Define the mapping F : R n Ñ R n by F pXq " X p1 `|X| 2 q 1{2 . Note that each u k satisfies divpF p∇u k qq " 0. We know that

F p∇u 1 q ´F p∇u 2 q " ż 1 0 d dt F pt∇u 1 `p1 ´tq∇u 2 qdt " ż 1 0 dF pt∇u 1 `p1 ´tq∇u 2 q ¨∇pu 1 ´u2 qdt " p ż 1 0
dF pt∇u 1 `p1 ´tq∇u 2 qdtq ¨∇pu 1 ´u2 q.

(3.2.7)

From this, we can conclude that v " u 1 ´u2 satisfies an equation of the form divpApxq∇vq " 0, where the matrix Apxq is given by Equation (3.2.7).

The remaining is to show that Equation (3.2.6) is a uniformly elliptic equation. Given a unit vector V P S n´1 Ă R n and X P R n , we see that

dF pXqV " V p1 `|X| 2 q 1{2 ´ă X, V ą p1 `|X| 2 q 3{2 X.
In particular, taking inner product with V gives

p1 `|X| 2 q 3{2 ă dF pXqV, V ą " p1 `|X| 2 q´ă X, V ą 2 ě p1 `|X| 2 q ´|X| 2 " 1.
It follows that Apxq is a weighted average of positive definite matrix and thus also a positive definite matrix.

Corollary 3.2.3. Let Ω Ă R n be an open connected neighborhood of the origin. If u 1 , u 2 : Ω Ñ R are two solutions to the minimal surface equation (See Equation (3.1.13)) with u 1 ď u 2 and u 1 p0q " u 2 p0q, then u 1 " u 2 .

Proof. Lemma 3.2.2 tells us that the difference v :" u 1 ´u2 satisfies divpa i,j ∇vq " 0, where the matrix a i,j is positive definite. We apply the strong maximum principle for the linear elliptic equation to v. (See [START_REF] Han | Elliptic partial differential equations[END_REF] or [Theorem 3.5, Page 48] of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]).

As in Lemma 3.2.1, a minimal hypersurface can be locally written as the graph of a solution to the minimal surface equation. As a consequence of Corollary 3.2.3 we know that

Corollary 3.2.4. If Σ 1 , Σ 2 Ă R n are two complete connected minimal hypersurfaces (without boundary), Σ 1 X Σ 2 ‰ H and Σ 2 lies on one side of Σ 1 , then Σ 1 " Σ 2 .

Monotonicity Formula and Area estimates.

Proposition 3.2.5. (The Monotonicity Formula) Suppose that Σ k Ă R n is a minimal submanifold and x 0 P Σ. Then for all 0 ă s ă t,

VolpBpx 0 , tq X Σq t k ´VolpBpx 0 , sq X Σq s k " ż Bpx 0 ,tqzBpx 0 ,sqXΣ |px ´x0 q N | 2 |x ´x0 | k`2 , (3.2.8)
where px ´x0 q N is the projection of the normal part of Σ of the vector px ´x0 q.

See [Chapter 3.2, Page 24-26] of [START_REF] Tobias | A course in minimal surfaces[END_REF] for a proof. As a consequence, we have Corollary 3.2.6. Suppose that Σ k Ă R n is a minimal submanifold and x 0 P R n . Then the function

Θ x 0 psq " Vol Bpx 0 , sq X Σq VolpBp0, sq Ă R k q is a non-decreasing function of s. Moreover, if x 0 P Σ, then Θ x 0 psq ě 1 and (3.2.9) VolpΣ X Bpx 0 , sqq ě ω k s k ,
where ω k is the volume of the unit ball Bp0, 1q Ă R n .

The area estimate (3.2.9) can be generalized to the Riemannian manifolds. Before we state it, we will recall the coarea formula. This formula asserts (See, for instant, Chapter 3.2 of [START_REF] Federer | Geometric measure theory[END_REF] for a proof) that if pM, gq is a Riemannian manifold and the function h : M Ñ R is a proper (i.e. h ´1pp8, tsq is compact for all t P R) Lipschitz function, then for any locally integral function f on M and t P R, (3.2.10)

ż hďt f |∇ M h| " ż t 8 ż h"τ f dτ.
Theorem 3.2.7. (See [Lemma 1, Page 445] of [START_REF] Meeks | Topology of three dimensional manifolds and the embedding problems in minimal surface theory[END_REF]) Let pM n`1 , gq be a Riemannian manifold whose sectional curvature is bounded by a positive constant K and Σ n a minimal submanifold. If for some point x 0 P Σ, d M px 0 , BM q and d M px 0 , BΣq are both greater than a constant ą 0, then for any δ ď mint , ipM qu

(3.2.11) VolpBpx 0 , δq X Σq ě C n K ż δ 0 t ´1psinpK tqq n dt,
where ipM q is the radius of injectivity of M and C n depends only on n.

Proof. Let rpxq be the distance function of M from x 0 to x P Σ. If rpxq is smaller than ipM q, the Hessian comparison theorem (See [Theorem 27, Page 175] of [START_REF] Petersen | Riemannian geometry[END_REF]) gives (3.2.12) Hesspe i , e i qprqpxq :" ∇ e i ∇ e i r ´∇∇e i e i prq ě K cotpKrq where x P Σ, te i u n i"1 is an orthonormal basis of T x Σ and N is a unit normal vector of Σ at x.

Summing Equation (3.2.12) and using the minimality of Σ, we have

∆ Σ r 2 pxq ě 2rpxq∆ Σ rpxq " 2rpxqp n ÿ i"1 ∇ e i ∇ e i r ´∇∇ Σ e i pe i q rq " 2rpxqp n ÿ i"1 Hesspe i , e i qprq `n ÿ i"1 ∇ ∇ N e i pe i q rq, " 2rpxqp n ÿ i"1
Hesspe i , e i qprq `∇Hpxq rq, because Hpxq "

n ÿ i"1 ∇ N e i e i " 2rpxq n ÿ i"
VolpΣ X Bpx 0 , δqq ě ż δ 0 VolpBpΣ X Bpx 0 , τ qqqdτ ě nC n K ´n ż δ 0 τ ´1psinpK τ qq n dτ.
(3.2.19)

Stable minimal surfaces

In this section, let us consider a stable minimal surface Σ n Ă pM n`1 , gq with trivial normal bundle. First, the stable condition is linked with the first eigenvalue of the operator L (See (3.1.21) or (3.1.22)). Second, if pM, gq has positive curvature, the stability condition gives several topological restrictions on Σ.

Eigenvalues and Stability.

The stability of a minimal surface is characterized by the first eigenvalue of L (See (3.1.21) or (3.1.22)). A minimal hypersurface Σ n Ă pM, gq is stable if for any smooth function η with compact support,

ż Σ ´ηLpηq " ż Σ |∇ Σ η| 2 ´RicpN, N qη 2 ´|A| 2 η 2 ě 0.
We see that the stable condition is equivalent to the first eigenvalue λ 1 pL, Ωq ě 0 for each Ω Ă Σ where

(3.3.1) λ 1 pL, Ωq :" inft ż Σ ´ηLpηq|η P W 1,2 0 pΩq and ż Σ η 2 " 1u.
By standard elliptic theory ( see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] or [START_REF] Han | Elliptic partial differential equations[END_REF]), we get the following:

Lemma 3.3.1. If L and Ω Ă Σ are assumed as above and λ 1 :" λ 1 pL, Ωq, then the eigenfunction u P W 1,2 0 pΩq of the first eigenvalue λ 1 ( i.e. Lpuq " ´λ1 u) is smooth. It follows from the regularity theory for elliptic equations (See [Theorem 8.14, Page 188] of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] or [START_REF] Han | Elliptic partial differential equations[END_REF]).

Together with the Harnack inequality, we see that any eigenfunction of the first eigenvalue can not change sign.

Lemma 3.3.2. Assume that u is a smooth function on Ω that vanishes on BΩ. If Lpuq " ´λ1 u where λ 1 " λ 1 pL, Ωq, then u can not change sign.

Proof. We may assume that u is not identically zero and ş Σ u 2 " 1. Since u vanishes on BΩ, so does |u|. In fact, |u| also achieve the minimum in (3. Proposition 3.3.3. Let Σ n Ă pM n`1 , gq be a minimal hypersurface with trivial normal bundle, L its stability operator (See Equation (3.1.21) or (3.1.22)), and Ω Ă Σ a bounded domain. If there exists a positive function u on Ω with Lpuq " 0, then Ω is a stable minimal surface.

Proof. Set qpxq " |A| 2 `Ric M pN, N q so that L " ∆ Σ `q. Since u ą 0, the function w :" log u is well-defined and satisfies

(3.3.2) ∆ Σ w " ´q ´|∇ Σ w| 2 .
Let us consider any compactly supported function f on Ω. Multiplying both sides of Equation (3.3.2) with f 2 and using integrating by parts give

ż Ω f 2 q `f 2 |∇ Σ w| 2 " ´żΩ f 2 ∆ Σ w ď 2 ż Σ f |∇ Σ f ||∇ Σ w| ď ż Σ f 2 |∇ Σ w| 2 `|∇ Σ f | 2 , (3.3.3)
where the second inequality follows from the Cauchy-Schwarz inequality. Canceling the ş Σ f 2 |∇ Σ w| 2 term, we have that ż

S

´f Lpf q ě 0 This finishes the proof.

We will give a characterization for a complete (non-compact) stable minimal hypersurfaces with trivial normal bundle. For such a hypersurface, the stability is equivalent to the existence of positive solution to the stability operator. (1) λ 1 pL, Ωq ě 0 for any bounded domain Ω Ă Σ;

(2) λ 1 pL, Ωq ą 0 for any bounded domain Ω Ă Σ;

(3) there is a positive function u over Σ with Lpuq " 0.

Proof. By Proposition 3.3.3, (3) implies (1). Clearly, (2) implies (1). To see the equivalence of ( 1) and (2), we consider any bounded domain Ω 0 and choose a strictly larger bounded domain Ω 1 . The variational characterization of the first eigenvalue (See (3.3.1)) implies that λ 1 pL, Ω 0 q ě λ 1 pL, Ω 1 q ě 0, where the second inequality follows from (1). Let u 0 be the first eigenfunction for L in Ω 0 . We define u 1 on Ω 1 by

u 1 pxq " $ & % u 0 pxq, if x P Ω 0 ; 0, otherwise.
where u 0 ě 0 and u 0 is not identically zero (See Lemma 3.3.2). If we had that λ 1 pL, Ω 0 q " 0, then the nonnegative function u 1 is an eigenfunction of λ 1 pL, Ω 1 q. Lemma 3.3.1 tells that Lpu 1 q " λ 1 pL, Ωqu 1 and u 1 is smooth. Since u 1 " 0 on Ω 1 zΩ 0 , the Harnack inequality (See [Theorem 8.20, Page 199] of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]) implies that u 1 is identically zero on Ω 1 . This is not possible. We can conclude that λ 1 pL, Ω 0 q ą 0. The equivalence of (1) and (2) follows.

The remaining is to show that (2) implies (3). To do this, fixed a point p P Σ and any r ą 0, let B Σ pp, rq be the geodesic ball in Σ with radius r and centered at p. Then the first eigenvalue λ 1 pL, B Σ pp, rqq is greater than zero. By Fredholm alternative (See [Theorem 6.15, Page 107] of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]), there exists a function v r satisfying:

(3.3.4) Lpv r q " ´|A| 2 ´RicpN, N q on B Σ pp, rq and v r " 0 on BB Σ pp, rq.

Setting u r " v r `1, Equation (3.3.4) gives (3.3.5) Lpu r q " 0 in B Σ pp, rq and u r " 1 on BB Σ pp, rq.

We claim that (3.3.6) u r ą 0 on B Σ pp, rq.

If u r is not a non-negative function, we choose a nonempty connected component Ω Ă Σ of the set tx P B Σ pp, rq|u r pxq ă 0u. Then, Lpu r q " 0 on Ω and u r " 0 on BΩ. This is to say, the first eigenvalue λ 1 pL, Ωq ď 0. This is in contradiction with (2). Therefore, we conclude that u r ě 0.

The claim follows from the Harnark inequality (See [Theorem 8.20, Page 199] of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]).

For each r, we define a positive function by w r pxq :" u r pxqpu r ppqq ´1 on B Σ pp, rq and see that Lpw r q " 0 and w r ppq " 1. Now, let us consider a compact set K Ă B Σ pp, R 0 q. We use the Harnack inequality (See [Theorem 8.20, Page 199] of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]) to have a positive constant Cpp, R 0 q, only depending on p and R 0 , satisfying for any r ą 2R 0 ,

(3.3.7) |w r pxq| ď Cpp, R 0 q if x P B Σ pp, R 0 q.
The interior Schauder estimate (See [Theorem 6.2, Page 90] of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]) gives that

|w r | C 2,α K ď C K
where the constant C K only depends on K and 0 ă α ă 1.

To sum up, we have a uniform C 2,α estimate for any w r where r ą 2R 0 . We use the Arzela-Ascoli theorem to extract a subsequence of w r that converges uniformly to w. This convergence ensures that Lpwq " 0 and w ě 0. The Harnack inequality (See [Theorem 8.20, Page 199] of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]) tells us that w is a positive function. This finishes the proof of the theorem.

Global Structure (I): the compact case.

Proposition 3.3.5. (See Page 8 of [START_REF] Schoen | On the structure of manifolds with positive scalar curvature[END_REF])Let Σ n Ă pM n`1 , gq be a compact stable minimal surface with trivial normal bundle. If pM, gq has positive scalar curvature, then pΣ, g Σ q is conformally equivalent to a metric of positive scalar curvature, where g Σ is the induced metric and n ą 2.

Proof. The stability of the minimal surface Σ and Equation (3.1.22) give

(3.3.8) ż Σ κ M η 2 ´κΣ η 2 `1{2|A| 2 η 2 ď ż Σ |∇ Σ η| 2
for any smooth function η on Σ.

Since κ M ą 0 on Σ, we can conclude that

(3.3.9) ´żΣ κ Σ η 2 ă ż Σ |∇ Σ η| 2
for all smooth function η.

Let λ be the first eigenvalue of the operator ∆ Σ ´n´2 2pn´1q κ Σ and u an eigenfunction of λ. That is,

(3.3.10) ∆ Σ u ´n ´2 4pn ´1q κ Σ u " ´λu.
From (3.3.9), we can conclude that λ ą 0. Otherwise (λ ď 0), multipling two sides of (3.3.10) by u and integrating, we see that ż

Σ |∇ Σ u| 2 " ´n ´2 4pn ´1q ż Σ κ Σ u 2 `λ ż Σ u 2 ď ´n ´2 2pn ´1q ż Σ κ Σ u 2 ď n ´2 2pn ´1q ż Σ |∇ Σ u| 2
where the last inequality follows from (3.3.9). This is impossible.

As the argument in Lemma 3.3.2, we have that the eigenfunction u is smooth and a positive function. Multiplying the metric g Σ by u 4 n´2 , under the new metric, the scalar curvature of Σ is

u ´n`2 n´2 pκ Σ ´4pn ´1q n ´2 ∆ Σ uq " 4pn ´1q n ´2 λu ´n`2 n´2 ą 0.
Next, we consider a stable minimal surface Σ Ă pM 3 , gq.

Corollary 3.3.6. (See [Theorem 5.1, Page 139] of [START_REF] Schoen | Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature[END_REF])Let Σ 2 Ă pM 3 , gq be a closed stable minimal surface with trivial normal bundle. If the complete manifold pM 3 , gq has positive scalar curvature, then Σ is S 2 or RP 2 and ż Σ κ M `1{2|A| 2 ď 4π.

Proof. Since Σ is compact and has no boundary, we choose the constant function η " 1 as a cut-off function. Equation (3.1.22) gives

(3.3.11) ż Σ κ M `1{2|A| 2 ď ż Σ K Σ " 2πχpΣq,
where the last equity follows from the Gauss-Bonnet formula. As a consequence, we see that χpΣq ą 0. That is to say, Σ is S 2 or RP 2 . Therefore, χpΣq ď 2. It gives the inequality in the assertion.

In the next part, we will use Cohn-Vesson's inequality [START_REF] Cohn-Vossen | Kürzeste Wege und Totalkrümmung auf Flächen[END_REF] to generalize (3.3.11) to the non-compact case.

Global Structure (II)

: the non-compact case. By Theorem 3.3.4, for a complete (non-compact) stable minimal surface Σ Ă pM 3 , gq, there is a positive function u over Σ with Lpuq " 0.

Theorem 3.3.7. ([Theorem 2, Page 126] in [START_REF] Fischer-Colbrie | On complete minimal surfaces with finite Morse index in three manifolds[END_REF]) Let Σ 2 Ă pM, gq be a complete (non-compact) stable minimal hypersurface. If the complete manifold pM 3 , gq has nonnegative scalar curvature (κpxq ě 0), then the new metric u 2 d 2 s is a complete metric on Σ with non-negative sectional curvature, where d 2 s is the induced metric and u is the positive function with Lpuq " 0 (as in Theorem 3.3.4).

Proof. By Theorem 3.3.4, there is a positive function u with Lpuq " 0. In addition, there exists a minimal geodesic ray γptq : r0, 8q Ñ Σ in the metric d2 s :" u 2 d 2 s, where t is arclength in the original metric d 2 s. It is obtained as below:

Fixed x P M and for any R ą 0, let us consider a geodesic ball B Σ px, Rq in the complete manifold pΣ, d 2 sq. Define u R :" u `ηR where η R is a smooth positive function satisfying

η R " $ & % 0, |x| ă R; 1, |x| ą 2R.
Then, since u R is bounded away from zero, the metric u 2 R d 2 s is complete. Therefore, there is a shortest geodesic γ R from x to BB Σ px, Rq in the metric u 2 R d 2 s. We can conclude that γ R must stay in B Σ px, Rq. (Otherwise, under the metric u 2 R d 2 s, there is another curve connecting x to BB Σ px, Rq whose length is shorter than γ R 's.)

Since u " u R on B Σ px, Rq, γ R is also a minimizing geodesic in d2 s. Each γ R can be parametrized with respect to arclength in the metric d 2 s. Let us consider the limit of these minimizing geodesics in pΣ, d2 sq. The sequence tγ R u sub-converges to a minimizing geodesic ray γptq that is parametrized by arclength in the metric d 2 s.

It remains to show the completeness of d2 s.

By the construction of γ, the completeness of d2 s will follow if we can show that γ has infinite length under the metric d2 s, i.e. it is sufficient to show that (3.3.12)

ż 8 0 upγptqqds " 8.
Since γ is a minimizing geodesic in d2 s " u 2 d 2 s, the second variation formula of acrlength gives (3.3.13) ż 8 0 pp dφ ds q 2 ´Kφ 2 q ds ě 0, where the smooth function φ has compact support in p0, 8q, dφ ds " u ´1 dφ ds , ds " uds and the sectional curvature K of pΣ, d2 sq is

(3.3.14) K " u ´2pK ´∆Σ log uq.
In addition, u is a positive function with Lpuq " 0. Namely,

Lpuq " ∆ Σ u ´Ku `pκpxq `1{2|A| 2 qu " 0
where ∆ Σ is the Laplace-Beltrami operator respect to pΣ, d 2 sq and K is the sectional curvature of pΣ, d 2 sq. Since κ ě 0, we see that ∆ Σ u ď Ku. In addition,

(3.3.15) ∆ Σ log u " u ´1∆ Σ u ´|∇ Σ u| 2 u 2 .
ż 8 0 pu 1 q 2 u 3 φ 2 ds ď ż 8 0 u ´1pK ´∆Σ log uqφ 2 ds ď ż 8 0 u ´1pφ 1 psqq 2 ds
where the first inequality follows from (3.3.14) and (3.3.15), the last inequality follows from (3.3.13) and upsq " upγpsqq.

We now show that ş 8 0 uds " 8. Take φ " uψ, where the smooth function ψ has compact support in p0, 8q. Then φ 1 " u 1 ψ `uψ 1 and u ´1pφ 1 q 2 " u ´1pu 1 q 2 ψ 2 `upψ 1 q 2 `2u 1 ψ 1 ψ.

The inequality (3.3.16) shows that

ż 8 0 pu 1 q 2 u 3 φ 2 ds " ż 8 0 pu 1 q 2 u ψ 2 ds ď ż 8 0 u ´1pu 1 q 2 ψ 2 `upψ 1 q 2 `2u 1 ψ 1 ψds. Therefore, 0 ď ż 8 0 upψ 1 q 2 `2u 1 ψψ 1 ds.
The integration by parts gives (3.3.17) 0 ď ż 8 0 ´upψ 1 q 2 ´2uψ 2 ψds.

Set ψpsq " sξpsq, where the smooth function ξ has compact support in r0, 8q. Then,

ψ 1 psq " ξpsq `sξ 1 psq, ψ 2 psq " sξ 2 psq `2ξ 1 psq.
Putting these equations into (3.3.17), we have

ż 8 0 ξ 2 uds ď ż 8 0 p´6sξξ 1 ´2s 2 ξξ 2 ´s2 pξ 1 q 2 quds.
Choose a smooth decreasing function ξ so that ξpsq " 1, for 0 ď s ď R, ξpsq " 0, for s ą 2R, and |ξ 1 | and |ξ 2 | are bounded by 2R ´1 and 4R ´2 respectively, for R ď s ď 2R. Then |sξ 1 psq| ď 4 and |sξ 2 psq| ď 16. We see that

ż R 0 uds ď ż 8 0 uξ 2 ds ď ż 8 0 p´6sξξ 1 ´2s 2 ξξ 2 ´s2 pξ 1 q 2 quds ď 72 ż 8 R uds.
This inequality implies that ş 8 0 uds " 8. That is to say, d2 s " u 2 d 2 s is a complete metric with nonnegative sectional curvature K ě 0. We apply the above argument to stable minimal surfaces in a 3-manifolds of uniformly positive scalar curvature (i.e. its scalar curvature is bounded away from zero). This result was firstly proved by Gromov-Lawson [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF] and Schoen-Yau in [START_REF] Schoen | The existence of a black hole due to condensation of matter[END_REF]. The generalization to the stable H-surface was due to H.Rosenberg [START_REF] Rosenberg | Constant mean curvature surfaces in homogeneously regular 3-manifolds[END_REF].

Proof. For any x P Σ, let R :" d Σ px, BΣq, where d Σ is the induced distance function in pΣ, d 2 sq. For any ą 0, the geodesic ball B Σ px, R ´ q in Σ, centered at x with radius R ´ , is contained in Int Σ.

It is sufficient to show that for any ą 0, R ´ ď 2π ? 3c . First, we solve the equation Lpuq " 0 on the B Σ px, R ´ q, where L is the stability operator (See Equation 3.1.22). As in the proof of Theorem 3.3.4, the first eigenvalue λ 1 pL, B Σ p0, R ´ qq ą 0.

Let us consider the Dirichlet problem:

(3.3.18)

$ & % Lpvq " K ´pκ `1{2|A| 2 q in B Σ p0, R ´ q v " 0 on BB Σ p0, R ´ q.
Since λ 1 pL, B Σ px, R ´ qq ą 0, we use Fredholm alternative (See [Theorem 6.15, Page 107] of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]) to find a solution v. Setting u " v `1. Equations (3.3.8) give that

Lpuq " 0 in B Σ px, R ´ q and u " 1 on BB Σ px, R ´ q. As in the proof of Theorem 3.3.4, the positivity of the operator L implies that u is a positive function.

Make a conformal change of the metric, d2 s " u 2 d 2 s on B Σ px, R ´ q. Let consider the minimizing geodesic γ from x to BB Σ px, R ´ q.

Let ã and a be the length of γ in the metrics d2 s and d 2 s, respectively. Note that R ´ ď a.

The second variation formula of arclength shows that

(3.3.19) ż ã 0 pp dφ ds q 2 ´Kφ 2 q ds ě 0,
where the smooth function φ has compact support in p0, aq, ds " uds and dφ ds " u ´1 dφ ds .

We have that

Lpuq " ∆ Σ u ´Ku `pκpxq `1{2|A| 2 qu " 0 K " u ´2pK ´∆Σ log uq.

where ∆ Σ is the Laplace-Beltrami operator in pΣ, d 2 sq, K and K are the sectional curvature of d2 s and d 2 s, respectively. Therefore,

(3.3.20) φ 2 Ku " φ 2 u 2 pKu ´∆Σ u `|∇ Σ u| 2 u q ě φ 2 u 2 pcu `|∇ Σ u| 2 u q
where the second inequality follows from Lpuq " 0 and κ ě c.

Putting (3.3.20) into (3.3.19), we see that (3.3.21) ż a 0 u ´1φ 2 pc `u´2 |u 1 | 2 qds ď ż a 0 u ´1p dφ ds q 2 ds.
where upsq " upγpsqq. Setting φ :" u 1{2 ψ, where ψ has compact support in p0, aq. We have

ż a 0 ψ 2 pc `u´2 |u 1 | 2 qds ď ż a 0 u ´1pu 1{2 ψ 1 `1{2u ´1{2 u 1 ψq 2 ds " ż a 0 pψ 1 q 2 `1{4p u 1 u q 2 ψ 2 `u´1 u 1 ψ 1 ψds, ď ż a 0 pψ 1 q 2 `p u 1 u q 2 ψ 2 `1{3pψ 1 q 2 ds.
where the last inequality follows from the Cauchy-Schwarz inequality (|u ´1u 1 ψ 1 ψ| ď 3{4p u 1 u q 2 ψ 2 `1{3pψ 1 q 2 ). Canceling the term p u 1 u q 2 ψ 2 on the two sides of the inequality, we have

ż a 0 ψ 2 ds ď 4 3c ż a 1 pψ 1 q 2 ds.
Choosing ψpsq " sinpπa ´1sq, we know that 1 ď 4 3c p π a q 2 . That is to say, a ď 2π ? 3c . Therefore, we see that for each ą 0,

R ´ ď a ď 2π ? 3c .
Namely, d Σ p0, BΣq " R ď 2π ? 3c . As a consequence, we have Corollary 3.3.9. In a complete Riemannian 3-manifold of uniformly positive scalar curvature, any orientable complete stable minimal surface is compact and homeomorphic to S 2 . Theorem 3.3.8 implies that a stable surface in a 3-manifold of uniformly positive scalar curvature is compact. By Corollary 3.3.6, it is homeomorphic to S 2 . Together with Cohn-Vesson's inequality [START_REF] Cohn-Vossen | Kürzeste Wege und Totalkrümmung auf Flächen[END_REF], we generalize Corollary 3.3.6 to the non-compact case and obtain the so-called extrinsic Cohn-Vesson inequality (See [Theorem 5.8, Page 18] of [START_REF] Wang | Contractible 3-manifolds and positive scalar curvatures (I)[END_REF]).

Theorem 3.3.10. (See Theorem 5.8 in [START_REF] Wang | Contractible 3-manifolds and positive scalar curvatures (I)[END_REF]) Let Σ 2 Ă pM 3 , gq be a complete (non-compact) immersed stable minimal surface. If the complete manifold pM 3 , gq has non-negative scalar curvature (κpxq ě 0), then

(3.3.22) ż Σ κpxq `1{2|A| 2 dv ď 2πχpΣq.
Moreover, if κ ą 0 and Σ is embedded, then Σ is properly embedded.

Proof. From Equation (3.1.22), the stability operator can be written as L :" ∆ Σ Ḱ `pκpxq `1{2|A| 2 q, where ∆ Σ is the Laplace-Beltrami operator of pΣ, d 2 sq. Since the non-compact surface Σ is stable minimal , we use Theorem 3.3.4 to find a positive fuction u with Lpuq " 0.

Consider the metric d2 s " u 2 d 2 s. Let K and dv be its sectional curvature and its volume form. We see that (3.3.23) K " u ´2pK ´∆Σ log uq and dv " u 2 dv.

Theorem 3.3.7 shows that pΣ, d2 sq is a complete surface with nonnegative sectional curvature K ě 0. We use the Cohn-Vossen inequality [START_REF] Cohn-Vossen | Kürzeste Wege und Totalkrümmung auf Flächen[END_REF] to have

(3.3.24) ż Σ K dv ď 2πχpΣq.
Since Lpuq " 0, then ş B Σ px,Rq u ´1Lpuqdv " 0, where B Σ px, Rq is the geodesic ball in Σ centered at x P Σ with radius R. We deduce that ż

B Σ p0,Rq κpxq `1 2 |A| 2 dv " ż B Σ p0,Rq pK ´u´1 ∆ Σ uqdv " ż B Σ p0,Rq K Σ ´p∆ Σ log u `u´2 |∇ Σ u|qdv ď ż B Σ p0,Rq u ´2pK ´∆Σ log uqu 2 dv " ż B Σ p0,Rqq K dv ď ż Σ K dv (3.3.25)
Putting (3.3.24) into (3.3.25) and taking R Ñ 8, we have that,

ż Σ κpxq `1{2|A| 2 dv ď 2πχpΣq.
Remark that since Σ admits a complete metric d2 s of nonnegative sectional curvature, we see χpΣq ď 1 (See details in Corollary 3.3.11).

In the following, we consider the case that Σ is embedded and κpxq ą 0. We have that

(3.3.26) ż Σ κpxqdv ď 2π.
Suppose that Σ is not proper. There is an accumulation point p of Σ so that the set Bpp, r{2q X Σ is a non-compact closed set in Σ. Namely, it is unbounded in pΣ, d 2 sq. Hence, there is a sequence tp k u of points in Bpp, r{2q X Σ going to infinity in pΣ, d 2 sq.

Therefore, we may assume that the geodesic discs B Σ pp k , r{2q in Σ are disjoint.

Define two constants K 0 :" sup xPBpp,rq |K M pxq| r 0 :" 1 2 mintr, i 0 , π ? K u where i 0 :" inf xPBpp,rq Inj M pxq and K M is the sectional curvature of pM, gq. The geodesic ball B Σ pp k , r 0 {2q in Σ is contained in Bpp, rq. Applying [Theorem 3, Appendix, Page 139] of [START_REF] Rosenvald | Stable complete surfaces with constant mean curvature[END_REF] to the geodesic disc B Σ pp k , r 0 {2q Ă Bpp, rq, we have AreapB Σ pp k , r 0 {2qq ě Cpr 0 , K 0 q. This leads to a contradiction as follows:

2π ě ż Σ κpxqdv ě ż Bpp,rqXΣ κpxqdv ě ÿ k ż B Σ pp k ,r 0 {2q κpxq ě inf xPBpp,rq κpxq ¨ÿ k AreapB Σ pp k , r 0 {2qq ě inf xPBpp,rq κpxq ¨ÿ k C " 8
Combining Theorem 3.3.7 and Theorem 3.3.10, we will give a new proof of the result of Theorem 2 in [START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF].

Corollary 3.3.11. (See [Theorem 2, Page 211] of [START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF]) Let Σ Ă pM, gq be an oriented complete non-compact stable minimal surface. If the complete manifold pM 3 , gq has nonnegative scalar curvature(κpxq ě 0), then Σ is diffeomorphic to R 2 or S 1 ˆR. If the latter case occurs, then Σ is totally geodesic and the scalar curvature κ of M is zero along Σ.

Moreover, if κpxq ą 0, then Σ is diffeomorphic to R 2 .

Proof. Since Σ is stable minimal, we use Theorem 3.3.4 to find a positive function u with Lpuq " 0, where L is the stability operator (See (3.1.21) and (3.1.22)). By Theorem 3.3.7, pΣ, u 2 d 2 sq is a complete 2-manifold with nonnegative sectional curvature, where d 2 s is the induced metric.

We apply the Soul theorem (See Theorem 1.11 and Theorem 2.1 in [START_REF] Cheeger | On the structure of complete manifolds of nonnegative curvature[END_REF]) to pΣ, u 2 d 2 sq. This theorem asserts that if pX, gq is a connected complete manifold with nonnegative sectional curvature, there is a compact totally convex, totally geodesic submanifold (called a soul of pX, gq) such that X is diffeomorphic to the normal bundle of the submanifold.

Therefore, there is a submanifold S Ă Σ (i.e. a soul) such that Σ is the normal bundle of S. In addition, since Σ is non-compact, we see that dimpSq ă dimpΣq (dimpSq " 0 or dimpSq " 1).

Case (I) If dimpSq " 0, S is a point. That is to say, Σ is diffeomorphic to R 2 . Case (II) If dimpSq " 1, S is homeomorphic to S 1 . Since Σ is oriented, the normal bundle is trivial. That is to say, Σ is homeomorphic to S 1 ˆR. We use Theorem 3.3.10 to see that ż Σ κpxq `1{2|A| 2 dv ď 2πχpΣq.

In this case, we see that χpΣq " 0. Therefore, Σ is totally geodesic and κ " 0 on Σ.

If κ ą 0, the latter case never occurs. We can conclude that Σ is diffeomorphic to R 2 .

Finally, we give an application of Lemma 3.3.8 in a complete 3-manifold.

Theorem 3.3.12 (See Theorem A and Theorem 1.1 in [START_REF] Wang | Simply connected open 3-manifolds with slow decay of positive scalar curvature[END_REF]). Assume that pM 3 , gq is a contractible complete 3-manifold. If there exists a number α P p´8, 2q such that lim inf rpxqÑ8 r α pxqκpxq ą 0,

where κpxq is the scalar curvature of pM, gq and rpxq is the distance function from some point 0 P M to x, then M 3 is diffeomorphic to R 3 .

Proof. From our assumption, there are two positive constants C and R 0 such that if rpxq ą R 0 , then

(3.3.27) κpxq ě C r α pxq . Claim: If R ą 2 max ! R 0 , p 4 1`α{2 π p3Cq 1{2 q 2 2´α
)

, then the induced map π 1 pM zBp0, 4Rqq Ñ π 1 pM zBp0, Rqq is trivial.

Suppose the contrary that there exists some R ą 2 max

! R 0 , p 4 1`α{2 π p3Cq 1{2 q 2 2´α
) so that the induced map π 1 pM zBp0, 4Rqq Ñ π 1 pM zBp0, Rqq is non-trivial. That is to say, there is a simple closed curve γ Ă M zBp0, 4Rq which is not contractible in M zBp0, Rq.

We use the work of Morrey [START_REF] Bradfield | Multiple integrals in the calculus of variations[END_REF][START_REF] Charles B Morrey | The problem of plateau on a riemannian manifold[END_REF] to find an area-minimizing disc Ω with boundary γ. The surface Ω intersects the set Bp0, Rq. Therefore, Ω X BBp0, 2Rq and Ω X BBp0, 4Rq are both nonempty.

Let us consider the set Σ :" Ω X pBp0, 4RqzBp0, 2Rqq. It is a stable minimal surface in pM, gq whose boundary is contained in the disjoint union of BBp0, 4Rq and BBp0, 2Rq. Since κpxq ě C p4Rq α on Bp0, 4RqzBp0, 2Rq, we use Lemma 3.3.8 to know that Σ is contained in the 2p4Rq α{2 π p3Cq 1{2 -neighborhood of BΣ. Since BΣ X BBp0, 2Rq and BΣ X BBp0, 4Rq are both nonempty, we see that

(3.3.28) 2R " dpBBp0, 2Rq, BBp0, 4Rqq ď 2 2p4Rq α{2 π p3Cq 1{2 . That is to say, R ă p 4 1`α{2 π p3Cq 1{2 q 2 2´α
. This is in contradiction of the choice of R. This finishes the proof of Claim.

From the above Claim, we see that M is simply-connected at infinity. From Remark 1.1.8, we know that M is diffeomorphic to R 3 . Corollary 3.3.13 (See Theorem 3.5 in [START_REF] Wang | Simply connected open 3-manifolds with slow decay of positive scalar curvature[END_REF]). Assume that pM 3 , gq is a simplyconnected open 3-manifold with π 2 pM q " Z. Let 0 P M be a point and rpxq a distance function from x to 0. If there exists a real number α P r0, 2q, such that,

lim inf rpxqÑ8 r α pxqκpxq ą 0, then M 3 is diffeomorphic to R ˆS2 .
See the proof in [START_REF] Wang | Simply connected open 3-manifolds with slow decay of positive scalar curvature[END_REF] CHAPTER 4

Convergence

In this chapter, we consider the convergence theory for minimal surfaces. First, we consider the convergence theory for minimal surface equations (See Equation (3.1.13)). Together with local properties of minimal surfaces, we discuss a classical theorem about convergence with finite multiplicity.

Then, we will introduce a new concept, the lamination. Our focus is the convergence theory for stable minimal laminations. Finally, we construct a family of stable minimal lamination in a complete contractible 3-manifold and discuss its limit.

The smooth convergence

Let u be a function from the unit disc B 2 to R. Its image in R 3 is a minimal surface if and only if

divp ∇u a 1 `|∇u| 2 q " 0.
Lemma 4.1.1. Let tf i u i be a sequence of functions from the unit disc B 2 to R. Each graph of f i in R 3 is minimal. If ||f i || C 2 ď C ă 8, then up to extracting a subsequence, f i converges smoothly on compact sets of B 2 to f and the graph of f is also a minimal surface.

Proof. By Arzelà-Ascoli Lemma, we may extract a subsequence so that f k converges to f in C 1,α -topology on a compact set for 0 ă α ă 1. According to the minimal surface equation (See Equation (3.1.13)) and the Schauder estimate for linear elliptic equation (See [Theorem 6.2, Page 90] of [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]), we see that this sequence converges in C m,αtopology on a compact set for each m. As a consequence, f also satisfies the minimal surface equation (see Equation (3.1.13)). That is to say, its graph is minimal. Definition 4.1.2. In a complete Riemannian 3-manifold pM, gq, a sequence tΣ n u of immersed minimal surfaces converges smoothly with finite multiplicity (at most m) to an immersed minimal surface Σ, if for each point p of Σ, there is a disc neighborhood D in Σ of p, an integer m and a neighborhood U of D in M (consisting of geodesics of M orthogonal to D and centered at the points of D) so that for n large enough, each Σ n intersects U in at most m connected components. Each component is a graph over D in the geodesic coordinates. Moreover, each component converges to D in C 2,α -topology as n goes to infinity.

Note that in the case that each Σ n is embedded, the surface Σ is also embedded. The multiplicity at p is equal to the number of connected component of Σ n X U for n large enough. It remains constant on each component of Σ.

Remark 4.1.3. Let us consider a family tΣ n u n of properly embedded minimal surfaces converging to the minimal surface Σ with finite multiplicity. Fix a compact simplyconnected subset D Ă Σ. Let U be the tubular neighborhood of D in M with radius and π : U Ñ D the projection from U onto D. It follows that the restriction π| ΣnXU : Σ n X U Ñ D is a m-sheeted covering map for small enough and n large enough, where m is the multiplicity.

Therefore, the restriction of π to each component of Σ n X U is also a covering map. Hence, since D is simply-connected, it is bijective. Therefore, each component of Σ n X U is a normal graph over D.

Let us recall a classical theorem about convergence with finite multiplicity.

Theorem 4.1.4. (See [Theorem 4.37, Page 49] of [START_REF] William H Meeks | The global theory of minimal surfaces in flat spaces[END_REF] or [Compactness Theorem, Page 96] of [START_REF] Michael T Anderson | Curvature estimates for minimal surfaces in 3-manifolds[END_REF]) Let tΣ k u kPN be a family of properly embedded minimal surfaces in a complete 3-manifold pM 3 , gq satisfying (1) each Σ k intersects a given compact set K 0 ; (2) for any compact set K in M , there are three constants C 1 " C 1 pKq ą 0, C 2 " C 2 pKq ą 0 and k 0 " k 0 pKq P N such that for each k ě k 0 , it holds that

(1) |A Σ k | 2 ď C 1 on K X Σ k , where |A Σ k | 2 is the square length of the second funda- mental form of Σ k , (2) AreapΣ k X Kq ď C 2 .
Then, after passing to a subsequence, Σ k converges to a properly embedded minimal surface with finite multiplicity in the C 8 -topology.

Proof. Choose a point p k P Σ k X K 0 . Extracting a subsequence, the sequence tp k u converges to some point p P M . We may suppose that the unit normal vector Ñ n Σ k | p k to the surface Σ k at p k converges to some unit vector in T p M . Namely, the tangent space T p k Σ k Ă T p k M converges to some plane T Ă T p M . From (1) and Lemma 3.2.1, we can express Σ k locally (near p) as some graphs of functions over T . That is to say, there is an open geodesic ball U centered at p such that for k large enough, ' each component of U X Σ k is the graph of some function on T . The Hessian of this function is bounded by the bound of the second fundamental form (See Lemma 3.2.1 and (1)); ' the number of components of U XΣ k is bounded by the area bound from Theorem 3.2.7 and (2);

We use Lemma 4.1.1 to extract a subsequence converging to a minimal surface. Then, a diagonal argument allows us find a subsequence converging smoothly to a minimal surface Σ. The area bound (See (2)) implies that this sequence converges with finite multiplicity.

From the strong maximal principle (See Corollary 3.2.1), the minimal surface Σ is embedded or self-intersects transversally. Since Σ k is embedded, we can conclude that Σ is also embedded.

In the following, we consider the convergence theory for stable minimal surfaces. For stable minimal surfaces in a 3-manifold, Schoen gave a uniform bound of the second fundamental form.

Lemma 4.1.5. (See [Theorem 3, Page 122] in [START_REF] Schoen | Estimates for stable minimal surfaces in three dimensional manifolds[END_REF]) Let Σ be an immersed stable surface in a Riemannian manifold pM 3 , gq. Given r 0 P p0, 1s, and a point p 0 P Σ, if Σ X Bpp 0 , r 0 q has compact closure in Σ, there are two constants 0 ă 0 ă 1 and C ą 0 depending only on the metric g near p 0 and the injective radius of pM, gq at p 0 so that

|A Σ | 2 ď Cr ´2
0 on Bpp 0 , 0 r 0 q X Σ. As a consequence, we have Corollary 4.1.6. Let pM, gq be a complete Riemannian manifold and tΣ k u a sequence of complete embedded stable minimal surfaces. If for any compact set K Ă M , there is a constant C 1 depending on K, satisfying that for each k

AreapΣ k X Kq ď C 1 ,
then after extracting a subsequence, Σ k converges smoothly to a complete stable minimal surface with finite multiplicity.

Minimal lamination

In the following,we assume that a complete contractible 3-manifold pM, gq is not homeomorphic to R 3 . As in Remark 1.1.13, M is an increasing union of closed handlebodies tN k u k satisfying that for each k, ' N k is homtopically trivial in N k`1 ; ' None of the N k is contained in a 3-ball. In addition, for each k, the genus of N k is greater than zero. (If not, there is some handlebody N k of genus zero, namely a 3-ball. That is to say, N 0 is contained in a 3-ball N k which is in contradiction with the last paragraph. ) 4.2.1. Construction of minimal laminations. From Lemma 1.2.7, each N k has a system of meridians tγ l k u gpN k q l"1 , where gpN k q is the genus of N k . Our target now is to construct a lamination L k :" Y l Ω l k Ă N k (i.e. a disjoint union of embedded surfaces) with BΩ l k " γ l k and "good" properties. Let us recall a result of Meeks and Yau (See Theorem 3.1.8 or [Theorem 6.28 Page 224] of [START_REF] Tobias | A course in minimal surfaces[END_REF]). It provides us a geometric version of loop theorem to construct them.

Theorem. (See [MIY80, MIY82], [Theorem 6.28 Page 224] of [START_REF] Tobias | A course in minimal surfaces[END_REF] or Theorem 3.1.8) Let pM 3 , gq be a compact Riemannian 3-manifold whose boundary is mean convex and γ a simple closed curve in BM which is null-homotopic in M . Then, γ bounds an area-minimizing disc and any such least area disc is properly embedded.

Remark 4.2 The boundary BM is mean convex. That is, BM is a piecewise smooth 2-manifold consisting of smooth surfaces tHu i . On each H i , the mean curvature is nonnegative.

Let Σ be an embedded area-minimizing disc with boundary γ. It intersects BM transversally. Therefore, Int Σ is contained in Int M .

Our strategy is to apply this theorem to pN k , g| N k q for each k. However, the boundary of N k may be not mean convex. To overcome it, we find a new metric g k on N k so that 1) pN k , g k q is a 3-manifold with mean convex boundary;

2)

g k | N k´1 " g| N k´1 .
The metric g k is constructed as below:

Let hptq be a positive smooth function on R so that hptq " 1, for any t P Rzr´ , s. Consider the function f pxq :" hpdpx, BN k qq and the metric g k :" f 2 g| N k . Under pN k , g k q, the mean curvature Ĥpxq of BN k is

Ĥpxq " h ´1p0qpH pxq `2h 1 p0qh ´1p0qq

Choosing ą 0 small enough and a function h with hp0q " 2 and h 1 p0q ą 2 max xPBN k |Hpxq|2 , one gets the metric g k which is the required candidate in the assertion.

In the following, we inductively construct the lamination L k Ă pN k , g k q.

When l " 1, there is an embedded area-minimizing disc Ω 1 k Ă pN k , g k q with boundary γ 1 k (See Theorem 3.1.8 or Theorem 6.28 of [START_REF] Tobias | A course in minimal surfaces[END_REF]). As in Remark 4.2, it intersects BN k transversally. Then, Int Ω 1 k Ă Int N k . Suppose that there are l disjointly embedded stable minimal discs tΩ i k u l i"1 with BΩ i k " γ i k .

Let us consider the Riemannian manifold pT k,l , g k | T k,l q, where T k,l :" N k z > l i"1 Ω l k . It is a handlebody of genus gpN k q ´l. For example, see the following figure.

Ω 1 k pN k , g k q γ 1 k pT k,1 , g k | T k,1 q Ω 1 k γ1 k ´Ω1 k γ1 k Figure 4.1.
The boundary of pT k,l , g k | T k,l q consists of two different parts. One is BN k z > l i"1 γ i l . The mean curvature is positive on this part. The other is 2l disjoint discs tΩ i k ´ul i"1 and tΩ i k `ul i"1 . The two discs Ω i k ´and Ω i k `are two sides of the same minimal disc Ω i k . The mean curvature vanishes on these discs.

Therefore, the boundary of pT k,l , g k | T k,l q is mean convex (See Remark 4.2). In addition, tγ i k u iąl is a system of meridian of the handlebody pT k,l , g k | T k,l q.

Then, we use Theorem 3.1.8 and the above remark to find an embedded stable minimal surface Ω l`1 k in the closure of pT k,l , g k | T k,l q with boundary γ l`1 k . From Remark 4.2, Ω l`1 k intersects the boundary of pT k,l , g k | T k,l q transversally. Hence, Int Ω l`1 k is contained in Int T k,l . That is to say, tΩ i k u l`1 i"1 are disjoint stable minimal surfaces for pN k , g k q.

This finishes the inductive construction.

To sum up, there exist gpN k q disjointly embedded meridian discs tΩ l k u. Define the lamination L k by > l Ω l k . It is a stable minimal lamination for the new metric g k and for the original one away from BN k (near N k´1 , for example).

The set L k XN k´1 is a stable minimal lamination in pM, gq. Each leaf has its boundary contained in BN k´1 . In addition, since N 0 is not contained in a 3-ball, We can conclude that each lamination L k intersects N 0 . The reason is below:

If the set L k X N 0 is empty, we choose a tubular neighborhood N pL k q in N k with small radius so that the set N pL k q X N 0 is also empty. That is to say, N 0 lies in the handlebody N k zN pL k q of genus zero (i.e. a 3-ball). This is in contradiction with our assumption that N 0 is not contained in a 3-ball. 4.2.2. Limits of minimal laminations. Let us consider the sequence tL k u k and its limit. From Lemma 4.1.5, we know that the sequence tL k u k satisfies the condition (1) in Theorem 4.1.4. However, it may not hold the condition (2) in Theorem 4.1.4. For example, in the Whitehead manifold, each N k is of genus one. The lamination L k is a meridian disc Ω 1 k Ă N k . From Theorem 2.1.2, Ω 1 k XIntN 1 has at least 2 k´1 components intersecting N 0 . We know that for k ą 1, each component pΣ, BΣq Ă pN 1 , BN 1 q of Ω 1 k X IntN 1 is a stable minimal surface in pM, gq.

Choose x 0 P Σ X N 0 and r 0 " 1 2 mintr, i 0 u, where r :" distpBN 0 , BN 1 q and i 0 :" inf xPN 1 Inj M pxq. We see that the ball Bpx 0 , r 0 q is in N 1 . We apply a result [Lemma 1, Page 445] in [START_REF] Meeks | Topology of three dimensional manifolds and the embedding problems in minimal surface theory[END_REF] to pN 1 , BN 1 q. Hence, it follows that AreapΣq ě AreapΣ X Bpx 0 , r 0 qq ě Cpi 0 , r 0 , Kq where K :" sup xPN 1 |K M pxq| and K M is the sectional curvature.

Therefore, one has that AreapN 1 X Ω 1 k q ě 2 k´1 C. The area of L k X N 1 goes to infinity as k goes to infinity. That is to say, the sequence tL k u k does not satisfy Condition (2).

Generally, the sequence tL k u k may not sub-converge with finite multiplicity. In the following, we consider the convergence toward a lamination. Definition 4.2.1. A codimension one lamination in a 3-manifold M 3 is a collection L of smooth disjoint surfaces (called leaves) such that Ť LPL L is closed in M 3 . Moreover, for each x P M there exists an open neighborhood U of x and a coordinate chart pU, Φq, with ΦpU q Ă R 3 so that in these coordinates the leaves in L pass through ΦpU q in slices form R 2 ˆttu X ΦpU q.

A minimal lamination is a lamination whose leaves are minimal. Finally, a sequence of laminations is said to converge if the corresponding coordinate maps converge in C 0,αtopology.

For example, R 2 ˆΛ is a lamination in R 3 , where Λ is a closed set in R. Note that any (compact) embedded surface (connected or not) is a lamination. In [Appendix B, Laminations] of [START_REF] Colding | The space of embedded minimal surfaces of fixed genus in a 3-manifold. iv: Locally simply connected[END_REF], Coding and Minicozzi describ the limit of laminations with uniformly bounded curvatures.

Proposition 4.2.2. (See Proposition B.1, Page 610] in [START_REF] Colding | The space of embedded minimal surfaces of fixed genus in a 3-manifold. iv: Locally simply connected[END_REF])Let M 3 be a fixed 3-manifold. If L i Ă Bpx, 2Rq Ă M is a sequence of minimal laminations with uniformly bounded curvatures (where each leaf has boundary contained in BBp0, 2Rq), then a subsequence, L j , converges in the C α -topology for any α ă 1 to a (Lipschitz) lamination L in Bpx, Rq with minimal leaves.

We use Proposition 4.2.2 to show that Theorem 4.2.3. The sequence tL k uof laminations sub-converges to a lamination L . Moreover, Each leaf in L is a complete minimal surface.

Proof. As constructed above, the intersection L j XN k is a stable minimal lamination or any j ą k. It may have many leaves (connected components). Each leaf has boundary contained in BN k . In addition, L j intersects N 0 .

From Lemma 4.1.5, there is a constant CpN k´1 q, depending on N k and g, so that for any j ą k, |A L j | 2 ď CpN k´1 q on N k´1 . Therefore, for j ą 3, tL j X N 2 u is a sequence of minimal laminations with uniformly bounded curvature where each leaf has boundary contained in BN 2 . We use Proposition 4.2.2 to extract a subsequence converging to a minimal lamination in N 1 . Each leaf has boundary contained in BN 1 .

We repeat the argument on each N k . A diagonal argument allows us to find a subsequence of tL k u converging to a lamination L . Each leaf is a complete minimal surface.

For our convenience, we may assume that the sequence L k converges to L . In the following, we will show that if pM, gq has positive scalar curvature, then each leaf in L is a (non-compact) stable minimal surface. Lemma 4.2.4. Let Σ be a compact minimal surface in a 3-manifold pX, gq (possibly with boundary) and p X, ĝq the double cover of pX, gq. The lift Σ of Σ is a connected minimal surface in p X, ĝq. Then Σ is stable minimal if an only if Σ is stable minimal.

Proof. Let L and L be the stable operators of Σ and Σk respectively. The operator p ˚pLq is equal to L, where p : Σ Ñ Σ is the double cover. Let λ 1 and λ1 be the first eigenvalues of L and L respectively.

It is sufficient to show that λ1 " λ 1 .

Let f be an eigenfunction for λ 1 (i.e. Lpf q " ´λ1 f ). The function f :" p ˚pf q satisfies that Lp f q " ´λ1 f . Hence, λ 1 ě λ1 .

Let τ be the desk transformation of the double cover p. The surface Σ is τ -invariant, (namely τ p Σq " Σ). The map τ is isometric on Σ.

Let ĥ be an eigenfunction for λ1 (i.e. Lp ĥq " ´λ 1 ĥ). We may assume that ĥ is τinvariant (τ p ĥq " ĥ). (If not, we replace it by τ p ĥq `ĥ.) The function ĥ is equal to p ˚phq, where h is a smooth function on Σ. We have that Lphq " ´λ 1 h. Therefore, λ1 ě λ 1 . We can conclude that λ1 " λ 1 .

Theorem 4.2.5. Each leaf in L is stable minimal.

Proof. Let L t be a leaf in the minimal lamination L . Case (I): If L t is a limit leaf (that is to say, the closure of L zL t contains L t ), we use the result of Meek, Pérez and Rosenberg (See [Theorem 1, Page 4] of [START_REF] William H Meeks | Limit leaves of a cmc lamination are stable[END_REF]) to have that it is stable minimal.

Case (II): If L t is not a limit leaf, the intersection L t X L zL t is empty. There is a tubular neighborhood N pL t q of L t such that the intersection N pL t q X L zL t is empty.

Let π be the projection from N pL t q to L t . For any point p and r ą 0, we consider the geodesic disc B Lt pp, rq in L t and the set N pp, rq :" π ´1pB Lt pp, rqq.

Choose one component Σ k of N pp, rq X L k for k ě 0. Since L k converges to L , the sequence tΣ k u k converges to some subset of L . In addition, the intersection L X N pp, rq has the unique component, B Lt pp, rq. Hence, Σ k converges to B Lt pp, rq.

We first consider the case when L t is 2-sided. In the following, we show that B Lt pp, rq is stable minimal for any r ą 0.

Step 1: Define the function d k : Σ k Ñ R. Let Ñ npxq be the unit normal vector to L t at x and π k :" π| Σ k . The map π k : Σ k Ñ B Lt pp, rq is a covering map for k large enough.

We define the function d k : Σ k Ñ R as follows:

d k pxq "ă exp ´1 π k pxq pxq, Ñ npπ k pxqq ą .
Step 2: π k is injective.

Recall that the area of L k is finite. The subset Σ k has a finite area. Therefore, π k is a finite cover for k large enough.

We argue it by contradiction. Suppose that π k is a m-sheeted covering (m ą 1). Let us consider three sets in Σ k as follows:

I T op :" x P Σ k |d k pxq " maxtd k px 1 q|x 1 P π ´1 k pπ k pxqu ( ; I M ed " x P Σ k | mintd k px 1 q|x 1 P π ´1 k pπ k pxqqu ă d k pxq ă maxtd k px 1 q|x 1 P π ´1 k pπ k pxqu ( ; I Bot " x P Σ k |d k pxq " mintd k px 1 q|x 1 P π ´1 k pπ k pxqqu ( .
From the homotopy lifting property of π k and m ą 1, these disjoint three set are open. In addition, Σ " I T op > I M ed > I Bot . Since Σ k is connected, there are at least two empty sets in these three sets.

However, since m ą 1, I T op and I Bot are two nonempty sets. This is in contradiction with the last paragraph. We conclude that π k is injective.

Step 3: The stability of B Lt px, rq.

As in

Step 2, Σ k can be written as the graph of some function f k over B Lt pp, rq for k large enough. The sequence tf k u converges in C 0,α -topology. Since Σ k is stable minimal, we use Lemma 4.1.5 and Lemma 4.1.1 to have that the sequence tf k u converges in C 2,αtopology. Namely, Σ k converges to B Lt pp, rq in C 2,α -topology. Therefore, B Lt pp, rq is stable minimal for any r ą 0.

Therefore, we can conclude that L t is stable minimal.

If L t is 1-sided, we consider the double cover { N pL t q of N pL t q and the lift Lt of L t . We choose r large enough so that B Lt pp, rq Ă L t is 1-sided. The lift { B Lt pp, rq of B Lt pp, rq is connected and 2-sided in the lift { N pp, rq of N pp, rq. Let Σk be one component of the pre-image of Σ k . It is a stable minimal surface. The reason is as follows:

The map Σk Ñ Σ k is a m 1 -sheeted cover map, where m 1 ď 2. If m 1 " 1, the map Σk Ñ Σ k is isometric. Therefore, Σk is stable minimal. If m 1 " 2, we use Lemma 4.2.4 to have that Σk is stable minimal.

As in

Step 1, we define the projection π : { N pL t q Ñ Lt and the function dk : Σk Ñ R. The sequence t Σk u converges to { B Lt pp, rq. The map πk :" π| Σk is a cover map for k large enough. Since AreapΣ k q ď AreapL k q ă 8 and Areap Σk q ď 2AreapΣ k q, Σk has a finite area. Therefore, πk is a finite cover for k large enlarge.

As in the above case, { B Lt pp, rq is stable minimal. From Lemma 4.2.4, B Lt pp, rq is stable minimal for r large enough. Namely, L t is stable minimal. Theorem 4.2.6. If pM, gq has positive scalar curvature, each leaf in L is noncompact.

Recall that a component Ω l k of L k is an area-minimizing disc with boundary BΩ l k in the closure of pT k,l´1 , g k q, where T k,l´1 " N k z > l´1 j"1 Ω j k and g k is obtained by modifying the metric g. In addition, g k | N k´1 is equal to g| N k´1 .

Proof. We argue by contradiction. Suppose that there exists a compact leaf L t in L .

Step 1: Topology of L t From the positivity of the scalar curvature, we use Corollary 3.3.6 to have that L t is a 2-sphere or a projective plane.

If L t is a projective plane, L t is 1-sided. Hence, M zL t is connected. There is a an embedded curve γ in M which intersects L t transversally at one point. The intersection numberof L t and γ is ˘1.

However, from the contractibility of M , γ is homotopically trivial. Hence, the intersection number of γ and L t is zero, a contradiction.

We conclude that L t is a 2-sphere.

Step 2: Area Estimate.

Since M is irreducible (See Remark 1.1.1), there is a 3-ball B Ă M with boundary L t . Let N 2 pBq be the tubular neighborhood of B with radius 2 . The set N 2 pL t q is a subset of N 2 pBq.

Since N 2 pBq is relatively compact, there is a positive integer k 0 , such that N 2 pBq Ă N k 0 ´1.

From now on, we fix the integer k ą k 0 . Let tΣ j k u j be the set of components of L k X N 2 pL t q. The component Σ j k is contained in some component Ω j k of L k :" > l Ω l k . In the following, we show that there is a constant C, independent of k and j, so that the area of Σ j k in pM, gq is less than C We may assume that Ω j k intersects BN 2 pBq transversally. The intersection Ω j k X BN 2 pBq :" tγ i u i has finitely many components. Each component γ i is a circle and bounds a unique closed disc D i in Ω j k . Since BN 2 pBq is a 2-sphere, there is an embedded disc D 1 i Ă BN 2 pBq with boundary γ i .

We claim that for any D i Ă Ω j k , AreapD i , g k q ď AreapD 1 i , g k q, where AreapD i , g k q is the area of D i in pN k , g k q.

We prove it by induction on j.

When j " 1, Ω 1 k is an area-minimizing disc in pN k , g k q. If the claim does not hold for some D i Ă Ω 1 k , we consider the disc pΩ 1 k zD i q Y γ i D 1 i with boundary BΩ 1 k . Its area is less than the area of Ω 1 k in pN k , g k q, a contradiction. Therefore, for any D i Ă Ω 1 k , we have AreapD i , g k q ď AreapD 1 i , g k q. We suppose that it holds for any l ď j and any D i Ă Ω l k . In the following, we consider that D i is contained in Ω j`1 k . If D 1 i X p> lďj Ω l k q is empty, this claim follows from the above argument.

If not, we may assume that

D 1 i intersects > lďj Ω l k transversally. The intersection D 1 i X Y lďj Ω l k :" tc m u cmPC i has finitely many components. Each component c m bounds a disc D 1 m Ă D 1 i . In addition, it also bounds a disc D m Ă > lďj Ω j k . Let C max i be the set of maximal circles of C i in D 1 i . These discs tD 1 m u cmPC max i are disjoint. The set D 1 i zpY cmPC max i D 1 m q is contained in N k z > lďj Ω l k . We consider the disc D 2 i :" D 1 i zpY cmPC max i D 1 m q Y cmPC max i pY cm D m q.
It is contained in the closure of pT k,j , g k q.

Since Ω j`1 k is an area-minimizing disc in the closure of pT k,j , g k q, we have that AreapD i , g k q ď AreapD 2 i , g k q.

If not, we consider the disc pΩ j`1 k zD i q Y γ i D 2 i . Its area is less than the area of Ω j`1 k , a contradiction.

From the inductive hypothesis, AreapD m , g k q ď AreapD 1 m g k q, for any c m . Hence,

AreapD 1 i , g k q " AreapD 1 i zpY cmPC max i D 1 m q, g k q `ÿ AreapD 1 m g k q ě AreapD 1 i zpY cmPC max i D 1
m q, g k q `ÿ AreapD m g k q " AreapD 2 i , g k q ě AreapD i , g k q. Therefore, we finish the proof of the claim.

We will show that the above claim implies an area estimate.

Let C max be the set of maximal circles of tγ i u i in Ω j k . We have that

Ω j k X N 2 pL t q Ă Ω j k X N 2 pBq Ă Y γ i PC max D i . Hence, Σ k
j is a subset of some D i . Recall that g k | N 2 pBq " g| N 2 pBq for k ą k 0 . For each k and j, we have that AreapΣ j k , g k q " AreapΣ j k , gq AreapBN 2 pBq, g k q " AreapBN 2 pBq, gq

We then have that AreapΣ j k , gq " AreapΣ j k , g k q ď AreapD i , g k q ď AreapD 1 i , g k q ď AreapBN 2 pBq, g k q " AreapBN 2 pBq, gq.

We conclude that for each k ą k 0 and j, AreapΣ j k , gq ď AreapBN 2 pBq, gq.

Step 3: Contradiction.

Choose a point p P L t and a point p k P L k X N pL t q so that lim kÑ8 p k " p. Let Σ j k k be the component of L k X N 2 pL t q passing through p k . As the proof in Step 2, we have that for k ě k 0 AreapΣ j k k , gq ď AreapBN 2 pBq, gq. From Lemma 4.1.5, the curvatures of these surfaces tΣ j k k u k are uniformly bounded in N 2 pL t q. By Theorem 4.1.4, the sequence tΣ j k k u sub-converges smoothly to a properly embedded surface Σ with finite multiplicity in N pL t q.

For our convenience, we assume that tΣ j k k u converges smoothly to Σ in N pL t q. The limit Σ Ă L is a disjoint union of connected embedded surfaces. Its boundary is contained in BN pL t q. In addition, p lies in L . Hence, L t is one component of Σ.

Since Σ is properly embedded, the set Σ 1 :" ΣzL t is a closed set. The sets Σ 1 and L t are two disjoint closed sets. Choose δ ă {2 small enough such that N 2δ pL t q X N 2δ pΣ 1 q " H.

Claim: For k large enough, Σ j k k is contained in N δ pL t q > pN 2 pL t qzN 2δ pL t qq. Since Σ j k k is a subset of N 2 pL t q, Σ j k k zN 2δ pL t q is contained in N 2 pL t qzN 2δ pL t q. It is sufficient to show that Σ j k k X N 2δ pL t q is contained in N δ pL t q. For k large enough, Σ j k k XN pL t q is contained in N δ pΣq, because Σ j k k XN pL t q converges to Σ. Hence, Σ j k k X N 2δ pL t q is a subset of N 2δ pL t q X N δ pΣq. From the choice of δ, we have that a) N δ pΣq is equal to N δ pL t q > N δ pΣ 1 q; b) N δ pΣ 1 q X N 2δ pL t q is empty.

By a), N 2δ pL t q X N δ pΣq is equal to N δ pL t q > pN δ pΣ 1 q X N 2δ pL t qq. From b), it is equal to N δ pL t q. Therefore, Σ k j k X N 2δ pL t q is contained in N δ pL t q. This finishes the proof of the claim.

For k large enough, p k is located in N δ pL t q. Namely, Σ j k k X N δ pL t q is non-empty. In addition, since BΣ j k k Ă BN 2 pL t q is non-empty, Σ j k k X pN 2 pL t qzN 2δ pL t qq is also nonempty. The sets N δ pL t q and N 2 pL t qzN 2δ pL t q are disjoint. Since Σ j k k is connected, we use the claim to have that one of these two sets Σ j k k X N δ pL t q and Σ j k k X pN 2 pL t qzN 2δ pL t qq is empty. This is in contradiction with the last paragraph.

We can conclude that each leaf L t is non-compact.

As an consequence, we give a new proof of [Corollary 10.8, Page 173] in [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF].

Corollary 4.2.7. A complete contractible 3-manifold with uniformly positive scalar curvature (i.e. that is, its scalar curvature is bounded away from zero) is homeomorphic to R 3 Proof. Suppose that M is not diffeomorphic to R 3 . As described above, there exists a complete (non-compact) stable minimal surface Σ. By Corollary 3.3.11, Σ is conformally diffeomorphic to R 2 .

Since the scalar curvature κpxq of M is uniformly positive, inf xPM κpxq ą 0. From Theorem 3.3.10, one has,

2π ě ż Σ κpxqdv ě inf xPM κpxq ¨żΣ dv " inf xPM κpxq ¨AreapΣq.
Therefore, Σ is a surface of finite area. However, we apply the theorem of Gromov and Lawson [Theorem 8.8] in [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF]. This theorem asserts that if pX, gq is a Riemannian manifold of positive scalar curvature, then any complete stable minimal surface of finite area in X is homeomorphic to S 2 . Hence, Σ is homeomorphic to S 2 , which leads to a contradiction with the topological structure of Σ (Σ is homeomorphic to R 2 ).

Part 3

Proof of Main Theorems CHAPTER 5

The Vanishing Property

In this chapter, we consider the geometry of a complete stable minimal surface and its relationship with the topological property of contractible 3-manifolds.

Let us consider a complete contractible Riemannian 3-manifold pM, gq of positive scalar curvature and a complete (non-compact) embedded stable minimal surface Σ Ă pM, gq. From Theorem 3.3.10 and Corollary 3.3.11, the surface Σ is a properly embedded plane (i.e. it is diffeomorphic to R 2 ).

In the following, we assume that M :" Y k N k is not homeomorphic to R 3 where tN k u is assumed as in Remark 1.1.13. By Theorem 2.2.6, there is an increasing family tR k u k of closed handlebodies with Property H. Definition. A complete embedded stable minimal surface Σ Ă pM, gq is called to satisfy the Vanishing Property for tR k u k , if there exists a positive integer kpΣq so that for any k ě kpΣq, any circle in Σ X BR k is contractible in BR k .

Let us consider a stable minimal lamination L Ă pM, gq, where each leaf is a complete (non-compact) stable minimal surface. It is called to have the Vanishing Property for tR k u k , if there is a positive integer k 0 so that for any k ě k 0 and each leaf L t in L, then any circle in L t X BR k is contractible in BR k .

We will consider the Vanishing property and its relationship with Property P and the fundamental group at infinity.

The vanishing Property and Property P

In this section, we consider the case that a complete contractible genus one 3-manifold pM, gq. In this case, we see from Lemma 1.3.10 that the family tN k u (as in Theorem 1.3.13) satisfies Property H. Namely, R k is defined as N k .

In addition, we see from Theorem 2.1.6 that the manifold M satisfies Property P (See Definition 2.1.3). If pM, gq has positive scalar curvature, the geometry of a stable minimal lamination is constrained by the extrinsic Cohn-Vesson inequality ( See Theorem 3.3.10) as well as by Property P. Their relationship is clarified by the following theorem: Theorem 5.1.1. Let L :" Y tPΓ L t be a stable minimal lamination in a complete contractible genus one 3-manifold pM, gq. Each leaf L t is a complete (non-compact) stable minimal lamination. If the manifold pM, gq has positive scalar curvature (κpxq ą 0), then L satisfies the Vanishing property for tN k u k , where tN k u k is assumed as in Theorem 1.3.13.

Precisely, there exists a positive integer k 0 " k 0 pM, gq, such that for each k ě k 0 and any t P Λ, each embedded circle γ in L t X BN k is contractible in BN k .

Proof. Since pM, gq has positive scalar curvature, we know from Corollary 3.3.11 that each L t is diffeomorphic to R 2 .

We prove by contradiction. We suppose that there exists a sequence of increasing integers tk n u n such that :

for each k n , there exists a minimal surface L tn in L and an embedded circle c kn Ă L tn X BN kn which is not contractible in BN kn .

Since lim nÑ8 k n " 8, we know that lim nÑ8 IpN 1 , N kn q " 8.

Because L tn is homeomorphic to R 2 , there exists a unique disc D n Ă L tn with boundary c kn . From Property P (Definition 2.1.3), we see that D n X Int N 1 has at least IpN 1 , N kn q components intersecting N 0 , denoted by tΣ j u m j"1 . Define the constants r :" d M pBN 0 , BN 1 q, C :" inf xPN 1 κpxq, K :" sup xPN 1 |K M | and i 0 :" inf xPN 1 pInj M pxqq , where K M is the sectional curvature of pM, gq and Inj M pxq is the injective radius at x of pM, gq.

Choose r 0 " 1 2 minti 0 , ru and x j P Σ j X N 0 , then Bpx j , r 0 q is in N 1 . We apply Theorem 3.2.7 to the minimal surface pΣ j , BΣ j q Ă pN 1 , BN 1 q. Hence, one has that AreapΣ j X Bpx j , r 0 qq ě C 1 pK, i 0 , r 0 q. From Theorem 3.3.10, we have:

2π ě ż Lt n κpxqdv ě m ÿ j"1 ż Σ j κpxqdv ě m ÿ j"1 ż Σ j XBpx j ,r 0 q κpxqdv ě m ÿ j"1 CAreapΣ j X Bpx j , r 0 qq ě CC 1 m ě CC 1 IpN 1 , N kn q
This contradicts the fact that lim nÑ8 IpN 1 , N kn q " 8 and completes the proof.

Remark 5.1.2. In the following, our proof requires that BN k intersects some leaf L t transversally. To overcome it , we will deform the solid torus N k in a small tubular neighborhood of BN k so that the boundary of the new solid torus intersects L t transversally.

This new solid torus also holds for Theorem 5.1.1. The reason is as follows:

The proof of Theorem 5.1.1 only depends on the extrinsic Cohn-Vossen inequality (See Theorem 3.3.8) and the geometric indexes. If we replace N k by a new solid torus obtained from deforming N k , all geometric indexes remain unchanged. Therefore, N 1 k also holds for Theorem 5.1.1.

Remark 5.1.3. Let tR k u k be a family of solid tori in a complete contractible 3manifold pM, gq with the properties that lim kÑ8 IpR 1 , R k q " 8. Now we consider the case that the maps π 1 pBR k q Ñ π 1 pM zR k q and π 1 pBR k q Ñ π 1 pR k zR 0 q are both injective. From Remark 2.1.7, the family satisfies Property P (See the detail in Remark 2.1.7).

Because Y k R k may not equal to M , the sequence tγ n u nąN of circles may not go to infinity. For overcoming it, we choose a new family tγ 1 n u nąN of circles going to infinity to replace it.

The map π 1 pBR kn X N j kn q Ñ π 1 pBR kn q is surjective (See Theorem 2.2.6 and Definition 2.2.5). Hence, we can find a circle γ 1 n Ă BN j kn X BR kn which is homotopic to γ n in BR kn . The sequence of circles tγ 1 n u něN goes to infinity. The sequence tγ 1 n u also have the property that for n ą N , ' γ 1 n is homotopic to γ 1 n`1 in M zR 0 ; ' γ 1 n is not nullhomotopic in M zR 0 . From Remark 1.1.10, π 8 1 pM q is not trivial. As a corollary, we have Corollary 5.2.2. Let pM, gq be a Riemannian 3-manifold of positive scalar curvature and tR k u k a family of handlebodies with Property (H). If π 8 1 pM q is trivial, then any complete stable minimal surface in pM, gq has the Vanishing property for tR k u k .

Theorem 5.2.3. Let pM, gq be a Riemannian manifold of positive scalar curvature and a family of handlebodies tR k u k with Property (H). If each leaf in a lamination L is a complete (non-compact) stable minimal surface satisfying the Vanishing Property for tR k u k , then the lamination L also has the Vanishing property for tR k u k .

Proof. We argue by contradiction. Suppose that there exists a sequence tL tn u of leaves in L and a sequence of increasing integers tk n u n so that some circle γ n Ă L tn XBR kn is not contractible in BR kn for each n.

The leaf L tn is a complete (non-compact) stable minimal surface. From Corollary 3.3.11 (See [Theorem 2, Page 211] of [START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF]), it is diffeomorphic to R 2 . The circle γ n bounds a unique closed disc D n Ă L tn . Since γ n is not nullhomotopic in M zR 0 (See Remark 2.2.7), the disc D n intersects R 0 .

Step 1: The sequence tL tn u n sub-converges with finite multiplicity.

Since each L tn is a stable minimal surface, we use Lemma 4.1.5 (See [Theorem 3, Page 122] of [START_REF] Schoen | Estimates for stable minimal surfaces in three dimensional manifolds[END_REF]) to show that, fixed a compact set K Ă M , there exists a constant C 1 " C 1 pK, M, gq satisfying that

|A Lt n | 2 ď C 1 on K X L tn
where |A Lt n | 2 is the squared norm of the second fundamental form of L tn .

From Theorem 3.3.10, ş

Lt n κdv ď 2π. Hence, AreapK X L tn q ď 2πp inf xPK κpxqq ´1.

From Theorem 4.1.4, the sequence tL tn u n sub-converges to a sublamination L 1 of L with finite multiplicity. In addition, L 1 is proper embedded. The lamination L 1 may has infinitely many components. Let L 2 Ă L 1 be a set of leaves intersecting R 0 . Since L 2 is properly embedded, L 2 has finitely many leaves.

Since each leaf L t in L 1 is homeomorphic to R 2 (See Corollary 3.3.11), any embedded circle γ Ă BR k X L t bounds a unique closed disc D Ă L t for k ą 0.

If L t is in L 1 zL 2 , the intersection D X R 0 is empty. Namely, γ is contractible in M zR 0 . Since π 1 pBR k q Ñ π 1 pM zR 0 q is injective, then γ is nullhomotopic in BR k .
Therefore, we conclude that for each k ą 0 and any leaf L t P L 1 zL 2 , any circle in L t X BR k is homotopically trivial in BR k .

Step 2: The Vanishing property gives a contradiction.

From now on, we abuse the notation and write tL tn u for a convergent sequence. In addition, we assume that the lamination L 2 :" > m s L ts (L 2 has finitely many leaves). The Vanishing property gives an integer kpL ts q for L ts . For k ě ř m s"1 kpL ts q, any circle in BR k X L 2 is contractible in BR k . From the above fact, for k ą 0, any closed curve in BR k X L 1 zL 2 is also homotopically trivial in BR k .

Therefore, for any k ě ř m s"1 kpL ts q, any circle in BR k X L 1 is contractible in BR k . In the following, we fix the integer k ě ř m s"1 kpL ts q and have the following: Claim: For n large enough, any circle in BR k X L tn is homotopically trivial we may assume that L 1 intersects BR k transversally. Since L 1 is properly embedded, BR k X L 1 has finitely many components. Each component of BR k X L 1 is an embedded circle. From the above fact, it is homotopically trivial in BR k . That is to say,

π 1 pBR k X L 1 q Ñ π 1 pBR k q is a trivial map. Choose an open tubular neighborhood U of L 1 X BR k in BR k . It is homotopic to L 1 X BR k in BR k . Therefore, π 1 pU q Ñ π 1 pBR k q is a trivial map.
Since tL tn u converges to L 1 , we see that L tn XBR k is contained in U for n large enough. Hence, the map π 1 pBR k X L tn q Ñ π 1 pBR k q is trivial. Namely, any circle in BR k X L tn is contractible in BR k . The claim follows.

The boundary γ n Ă BR kn of D n is non-contractible in BR kn . From Remark 2.2.7, it is non-contractible in M zR 0 . If k n ą k, we use Corollary 1.2.6 and Property H (See Definition 2.2.5) to find a meridian γ 1 Ă D n Ă L tn of R k . This is in contradiction with the above claim.

As a consequence, we have Corollary 5.2.4. Let pM, gq be a Riemannian manifold of positive scalar curvature and tR k u k a family of handlebodies with Property (H). If π 8 1 pM q is trivial, then any complete stable minimal lamination in pM, gq has the Vanishing property for tR k u k .

CHAPTER 6

Proof of Main Theorems

In this chapter, we will explain the proof of the main theorems. For a contractible 3manifold, the existence of complete metrics of positive scalar curvature and its topological properties (for example, Property H) can be related through the limit of a sequence of lamination (constructed in Chapter 4). Combining all these, we will finish the proof of Theorem B 2 and Theorem C.

Proof of Main theorems

For the proof of the main theorems, we will argue by contradiction. In this chapter, we assume that pM, gq is a complete contractible 3-manifold of positive scalar curvature which is not homeomorphic to R 3 .

As in Remark 1.1.13, M is an increasing union of handlebodies tN k u k with the property that for each k, (1) N k is homotopically trivial in N k`1 ; (2) none of the N k is contained in a 3-ball. In addition, the genus of N k is greater than zero for k ě 0.

From Lemma 1.2.7, each N k has a system of meridians tγ l k u gpN k q l"1 . As in Chapter 4.2, there is a lamination L k :" > l Ω l k Ă N k . Each leaf Ω l k is a meridian disc with boundary γ l k . As described in Chapter 4.2, since N 0 is not contained in a 3-ball, the lamination L k intersects N 0 .

The intersection L k X N k´1 is a stable minimal lamination in pM, gq. From Chapter 4.2, the sequence tL k u sub-converges to a stable minimal lamination L :" Y tPΓ L t in pM, gq. Each leaf L t is a complete (non-compact) stable minimal surface in pM, gq.

Since pM, gq has positive scalar curvature, each leaf in L is a properly embedded plane (See Theorem 3.3.10 and Corollary 3.3.11).

6.1.1. Properties of L . In the following, we consider the lamination L and its properties.

If one of the following holds:

' π 8 1 pM q is trivial, ' M is a contractible genus one 3-manifold; ' M is homeomorphic to M 1 (constructed in Chapter 1.3.5) there is a family of ascending handlebodies tR k u k satisfying Property H, so that a) the lamination L has the Vanishing property for tR k u k ; b) for each k and any N j containing R k , the intersection L j X BR k has at least one meridian of R k . Remark 6.1.1.

' The three different conditions for M correspond respectively to Theorem C, Theorem B 2 and Theorem 1.3.15. ' If M is a contractible genus one 3-manifold, then π 8 1 pM q is trivial. In the following, let us explain the above properties, a) and b).

If π 8

1 pM q is trivial, we know from Theorem 2.2.6 that there is an increasing family tR k u k of closed handlebodies with Property H (See Definition 2.2.5). Corollary 5.2.4 shows that the lamination L has the Vanishing Property for this family.

In addition, none of the R k is contained in a 3-ball (See Definition 2.2.5). Together with Property H, we use Corollary 1.2.8 to know that if N j contains R k , the intersection L j X BR k has at least one meridian of R k .

If M is a contractible genus one 3-manifold, the family tN k u can be assumed as in Theorem 1.3.13. That is to say, the geometric index IpN k , N k`1 q is greater than zero.

From Lemma 1.3.10, Theorem 1.1.4 and Lemma 1.1.5, we know that the map π 1 pBN k q Ñ π 1 pM zN k q and π 1 pBN k q Ñ π 1 pN k zN 0 q are also injective. That is to say, the family tN k u satisfies Property H.

From Theorem 2.1.6, M satisfies Property P. Theorem 5.1.1 implies that L satisfies the Vanishing property for tN k u k . In addition, since the geometric index IpN j , N k q is greater than zero for j ă k, L j X BN j has at least one meridian of N j (See Lemma 1.3.11).

If M is homeomorphic to M 1 ( constructed in Chapter 1.3.5), then there is a properly embedded plane P 1 as constructed in Chapter 1.3.4. It cuts M 1 into two contractible 3-manifolds, M 1 1 and M 2 1 (See Proposition 1.3.14). In addition, M 1 1 is homeomorphic to the Whitehead manifold. As in the construction of the Whitehead manifold (in Chapter 1.3.2) there is a family tR k u of solid tori in M satisfying:

' the union Y k R k is equal to M 1 1 ; ' the geometric index IpR k , R k`1 q is equal to two for each k.
As in the genus one case, we know that π 1 pBR k q Ñ π 1 pM zR k q and π 1 pBR k q Ñ π 1 pR k zR 0 q are both injective (See Lemma 1.3.10). In addition, none of the R k is contained in a 3-ball in M . That is to say, the family tR k u k satisfies Property H.

From Remark 2.1.7, the family tR k u k satisfies Property P. Since lim kÑ8 IpR 0 , R k q " lim kÑ8 2 k´1 " 8, we see from Remark 5.1.3 that L satisfies the Vanishing property for tR k u.

In addition, we know from Corollary 1.2.8, that for any N k containing R j , L k has at least one meridian of R j for k ě j. Remark 6.1.2. In the following, our proof requires that BR k intersects some leaf L t transversally. To overcome it , we will deform the handlebody R k in a small tubular neighborhood of BR k so that the boundary of the new handlebody intersects L t transversally.

This new handlebody also satisfies a) and b). The reason is as follows: For any handlebody R 1 k obtained by deforming R k , the maps π 1 pBR 1 k q Ñ π 1 pR 1 k zR 0 q and π 1 pBR 1 k q Ñ π 1 pM zR 1 k q are both injective. The proof of a) and b) just depends on the injectivity of these two maps. Hence, the handlebody R 1 k also holds a) and b).

Remark. The positivity of the scalar curvature gives the property, a). The topological properties of M implies that L k satisfies the property, b).

In the following, we just give the complete proof of Theorem C. The remaining proofs of Theorem B 2 and Theorem 1.3.15 are the same as Theorem C.

6.1.2. The proof of Theorem C. From the above property a), there is a family of handlebodies tR k u k so that L has the Vanishing property for the family tR k u k . That is to say, There is a integer k 0 ą 0 so that for any k ě k 0 , any circle in L X BR k is contractible in BR k .

If L :" > tPΛ L t has finitely many components, we may assume that each leaf L t intersects BR k transversally for k ě k 0 . Since L t is properly embedded, L t XBR k :" tγ t i u iPIt has finitely many components. Each component is a circle.

From the above fact, each γ t i is null-homotopic in BR k . Consider the unique closed disc D t i Ă BR k with boundary γ t i and the partially ordered set ptD t i u tPΛ,iPCt , Ăq. Let C be the set of maximal elements. In particular, it is a finite set. The set L X BR k is contained in the disjoint union of closed discs in C.

In the general case, we also have a similar result.

Lemma 6.1.3. For any k ě k 0 , BR k p q X L is contained in a disjoint union of finitely many closed discs in BR k p q, where R k p q :" R k zN pBR k q, N pBR k q is some tubular neighborhood of BR k in R k .

In the general case when L has infinitely many components, we will prove it in Chapter 6.3.

We now finish the proof of Theorem C.

Proof. Suppose that some complete contractible 3-manifold pM, gq with positive scalar curvature and trivial π 8 1 pM q is not homeomorphic to R 3 . As above, there is an ascending family tR k u k of handlebodies with Property H, so that a) the lamination L has the Vanishing property for tR k u k ; b) for each k and any N j containing R k p q, the intersection L j X BR k p q has at least one meridian of R k p q.

The Vanishing property implies Lemma 6.1.3 (We will prove it in Chapter 6.3). That is to say, the intersection L X BR k p q is in the union of disjoint closed discs tD i u s

i"1 for k ě k 0 . Choose an open neighborhood U of the closed set L X R k`1 so that U X BR k p q is contained in a disjoint union > s i"1 D 1 i , where D 1
i is an open tubular neighborhood of D i in BR k p q with small radius. Each D 1

i is an open disc in BR k p q. Since L k subconverges to L , there exists an integer j, large enough, satisfying

' L j X R k`1 Ă U ; ' R k p q is contained in N j . Therefore, L j X BR k p q is contained in U X BR k p q Ă >D 1 i . The induced map π 1 pL j X BR k p qq Ñ π 1 p> i D 1
i q Ñ π 1 pBR k p qq is a trivial map. We can conclude that any circle in L j X BR k p q is contractible in BR k p q.

However, from b), there exists a meridian γ Ă L j X BR k p q of R k p q. This contradicts the last paragraph and finishes the proof of Theorem C.

Two topological lemmas

Before proving Lemma 6.1.3, we introduce two topological lemmas. These two lemmas play a crucial role in the proof of Lemma 6.1.3 Lemma 6.2.1. Let pΩ, BΩq Ă pN, BN q be a 2-sided embedded disc with some closed subdiscs removed, where N is a closed handlebody of genus g ą 0. Each circle γ i is contractible in BN , where BΩ " > i γ i . Then N zΩ has two connected components. Moreover, there is a unique component B satisfying that the induced map π 1 pBq Ñ π 1 pN q is trivial.

Proof. We argue by contradiction. Suppose that N zΩ is path-connected. That is to say, there is an embedded circle σ Ă N which intersects Ω transversally at one point.

Since each γ i is contractible in BN , it bounds a unique disc D i Ă BN . The surface Ω :" Ω Ť i Y γ i D i also intersects σ transversally at one point. The intersection number between Ω and σ is ˘1.

However, Ω is the image of a continuous map g : S 2 Ñ N . It is contractible in N , since π 2 pN q " t0u. The intersection number between Ω and σ must be zero, which leads to a contradiction.

Therefore, N zΩ is not connected. Since Ω is 2-sided and connected, N zΩ just has two components B 0 and B 1 .

Remark that the surface Ω :" Ω Ť pY γ i D i q is an immersed 2-sphere in N . This deduces that the map π 1 pΩq Ñ π 1 p Ωq is trivial map. Therefore, the map π 1 pΩq Ñ π 1 pN q is trivial.

In the following, let us explain the existence of B. Consider the partially ordered relationship over tD i u induced by inclusion. Therefore, Y i D i is equals to a disjoint union of maximal elements in ptD i u, Ăq. The set BN z Y i D i is a compact surface with some disjoint closed sub-discs removed.

Therefore, the induced map π 1 pBN z Y i D i q Ñ π 1 pBN q is surjective. The induced map π 1 pBN q Ñ π 1 pN q is also surjective. We can conclude that the composition of these two maps π 1 pBN z Y i D i q Ñ π 1 pN q is also surjective.

The set BN zY i D i is contained in one of two components, B 1 and B 2 , of N zΩ. Without loss of generality, we may assume that B 1 contains BN zY i D i . Based on the last paragraph, the induced map π 1 pB 1 q Ñ π 1 pN q is surjective.

Let G i be the image of the map π 1 pB i q Ñ π 1 pN q, a subgroup of π 1 pN q. Van-Kampen's Theorem (See Theorem 1.1.4) gives an isomorphism between π 1 pN q and π 1 pB 1 q ˚π1 pΩq π 1 pB 2 q. Since the image of π 1 pΩq Ñ π 1 pN q is trivial, π 1 pN q is isomorphic to G 1 ˚G2 . Grushko's Theorem [START_REF] Aleksandrovich | On the bases of a free product of groups[END_REF] shows that rankpG 1 q `rankpG 2 q " rankpπ 1 pN qq. (The rank of a group is the smallest cardinality of a generating set for the group.) From the last paragraph, the image, G 1 , of the map π 1 pB 1 q Ñ π 1 pN q is isomorphic to π 1 pN q. That is to say, rankpG 1 q " rankpπ 1 pN qq. Therefore, rankpG 2 q is equal to zero. That is to say, G 2 is a trivial group. We know that B :" B 2 is the required candidate in the assertion.

Finally, we prove the uniqueness. Suppose that the two induced maps are both trivial. Therefore, the map H 1 pB i q Ñ H 1 pN q is trivial for each i. Applying the Mayer-Vietoris 6.3.1.1. Elements in S. Let tΣ t i u iPIt be the set of components of L t X Int R k for each t P Λ. (It may be empty.) We will show that for each component Σ t i , R k zΣ t i has a unique component B t i satisfying that π 1 pB t i q Ñ π 1 pR k q is trivial. If L t intersects BR k transversally, the boundary BΣ t i Ă L t X BR k is the union of some disjointly embedded circles. From the Vanishing property, any circles in the boundary BΣ t i Ă L t X BR k is contractible in BR k .

In addition, since L t is homeomorphic to R 2 and Σ t i is relatively compact, Σ t i is homeomorphic to an open disc with some disjoint closed subdiscs removed. By Lemma 6.2.1, R k zΣ t i has a unique component B t i satisfying that π 1 pB t i q Ñ π 1 pR k q is trivial. In general, L t may not intersect BR k transversally. To overcome it, we will deform the surface BR k . Precisely, for the leaf L t , there is a new handlebody Rk p t q containing R k so that L t intersects B Rp t q transversally, where Rk p t q is a closed tubular neighborhood of R k in M .

We consider the component Σt i of L t X Int Rk p t q containing Σ t i . As above, Rk p t qz Σt i has a unique component Bt i so that the map π 1 p Bt i q Ñ π 1 p Rk p t qq is trivial. Choose the component B t i of Bt i X R k whose boundary contains Σ t i . It is a component of R k zΣ t i . In addition, the map π 1 pB t i q Ñ π 1 p Bt i q Ñ π 1 p Rk p t qq is trivial. Since R k and Rk p t q are homotopy equivalent, the map π 1 pB t i q Ñ π 1 pR k q is also trivial. This finishes the construction of B t i . 6.3.1.2. Properties of S. From Lemma 6.2.2, for any B t i and B t 1 i 1 , it holds one of the following (1) B t i X B t 1 i 1 " H; (2) B t i Ă B t 1 i 1 ; (3) B t 1 i 1 Ă B t i , where t, t 1 P Λ, i P I t and i 1 P I t 1 .

Therefore, ptB t i u tPΛ,iPIt , Ăq is a partially ordered set. We consider the set tB j u jPJ of maximal elements. However this set may be infinite. Definition 6.3.1. S :" tB j |B j X R k p {2q ‰ H, for any j P Ju, where R k p {2q is R k zN {2 pBR k q and N {2 pBR k q is a 2-sided tubular neighborhood of BR k with radius {2. Proposition 6.3.2. Let Σ t i be one component of L t X R k and B t i assumed as above. If B t i is an element in S, then Σ t i X R k p {2q is nonempty.

Proof. We argue by contradiction. Suppose that Σ t i XR k p {2q is empty. As mentioned above, Σ t i cuts R k into two components. Hence, R k p {2q must be in one of these two components.

In addition, from the definition S, the component B t i of R k zΣ t i must intersect R k p {2q. One knows that R k p {2q is contained in B t i . However, the composition of maps π 1 pR k p qq Ñ π 1 pB t i q Ñ π 1 pR k q is an isomorphism. Therefore, the map π 1 pB t i q Ñ π 1 pR k q is non-trivial and surjective, which contradicts the fact that the map π 1 pB t i q Ñ π 1 pR k q is trivial. This finishes the proof.

Proposition 6.3.3. R k p q X L Ă Ť B j PS B j X R k p q. Moreover, BR k p q X L Ă Ť B j PS B j X BR k p q.

Proof. Each component Σ t i of L t X Int R k is contained in B t i . Hence, L t X R k is in Y iPIt B t i . We can conclude that L X R k is contained in YB t i . The set YB t i is equal to Y jPJ B j , because tB j u jPJ is the set of all maximal elements in the partially ordered set ptB t i u, Ăq. Therefore, L X R k is in Y jPJ B j . From the definition of S, Y jPJ B j X R k p q equals Y B j PS B j X R k p q. Therefore, R k p q X L Ă Y B j PS B j X R k p q.

Similarly, one has that Y jPJ B j XBR k p q equals Y B j PS B j XBR k p q. Hence, BR k p qXL Ă Y B j PS B j X BR k p q 6.3.2. The finiteness of the set S. The set BB j X Int R k equals some Σ t i Ă L t for t P Λ. Let us consider the set S t :" tB j P S|BB j X Int R k Ă L t u. Then, S " > tPΛ S t . Note that each B j P S t is a B t i for some i P I t . In this subsection, we first show that each S t is finite. Then, we argue that tS t u tPΛ contains at most finitely many nonempty sets. These imply the finiteness of S. Lemma 6.3.4. Each S t is finite.

Proof. We argue by contradiction. Suppose that S t is infinite for some t. For each B j P S t , there exists a i P I t so that B j is equal to B t i , where B t i is a component of R k zΣ t i and Σ t i is one component of L t X Int R k . By Proposition 6.3.2, Σ t i X R k p {2q is nonempty.

Choose x j P Σ t i X R k p {2q and r 0 " 1 2 mint {2, i 0 u, where i 0 :" inf xPR k Inj M pxq. Then the geodesic ball Bpx j , r 0 q in M is contained in R k .

We apply Theorem 3.2.7 to the minimal surface pΣ t i , BΣ t i q Ă pR k , BR k q. One knows that, AreapΣ t i X Bpx j , r 0 qq ě Cpr 0 , i 0 , Kq where K " sup xPR k |K M |. This leads to a contradiction from Theorem 3.3.10 as below: AreapBpx j , r 0 q X Σ t i q ě C inf

xPR k pκpxqq|S t | " 8
This finishes the proof.

Lemma 6.3.5. tS t u tPΛ contains at most finitely many nonempty sets.

Proof. We argue by contradiction. Suppose that there exists a sequence tS tn u nPN of nonempty sets. For an element B jt n P S tn , there is some i n P I tn so that B jn equals B tn in where B tn in is one component of R k zΣ tn in and Σ tn in is one of components of L tn X Int R k . Note that π 1 pB tn in q Ñ π 1 pR k q is trivial. By Proposition 6.3.2, Σ tn in X R k p {2q is not empty. Pick a point p tn in Σ tn in X R k p {2q.

Step1: tL tn u subconverges to a lamination L 1 Ă L with finite multiplicity.

Since L tn is a stable minimal surface, We know from Lemma 4.1.5 that for any compact set K Ă M , there is a constant C 1 :" C 1 pK, M, gq such that

|A Lt n | 2 ď C 1 on K X L tn .
From Theorem 3.3.10, ş

Lt n κpxqdv ď 2π. Hence, AreapK X L tn q ď 2πp inf xPK κpxqq ´1.

We use Theorem 4.1.4 (See [Compactness Theorem, Page 96] in [START_REF] Michael T Anderson | Curvature estimates for minimal surfaces in 3-manifolds[END_REF]) to find a sub-sequence of tL tn u subconverging to a properly embedded lamination L 1 with finite multiplicity. Since L is a closed set in M , L 1 Ă L is a sublamination.

From now on, we abuse notation and write tL tn u and tp tn u for the convergent subsequence. Step 2: tΣ tn in u converges with multiplicity one. Let L t8 be the unique component of L 1 passing through p 8 , where p 8 " lim nÑ8 p tn . The limit of tΣ tn in u is the component Σ 8 of L t8 X R k passing through p 8 , where Σ tn in is the unique component of R k X L tn passing though p tn .

Let D Ă L t8 be a simply-connected subset satisfying Σ 8 Ă D. Since tL tn u converges smoothly to L t8 , there exists 1 ą 0 and an integer N such that Σ tn in Ă Dp 1 q, for n ą N, where Dp 1 q is the tubular neighborhood of D with radius 1 in M . (See Definition 4.1.2 and Remark 4.1.3). Let π : Dp 1 q Ñ D be the projection. For n large enough, the restriction of π to each component of L tn X Dp 1 q is injective (See Remark 4.1.3).

Hence, π| Σ tn in : Σ tn in Ñ D is injective. That is to say, Σ tn in is a normal graph over a subset of D. Therefore, tΣ tn in u converges to Σ 8 with multiplicity one (See Definition 4.1.2). That is to say, there is a geodesic disc B Σ8 pp 8 q Ă Σ 8 centered at p 8 with small raduis so that p˚˚q: the set π ´1pB Σ8 pp 8 qq X Σ tn in is connected and a normal graph over B Σ8 pp 8 q, for large n.

Step 3: Get a contradiction.

There exists a neighborhood U of p 8 and a coordinate map Φ, such that each component of ΦpL X U q is R 2 ˆtxu X ΦpU q for some x P R. (See Definition 4.2.1 or Appendix B of [START_REF] Tobias | A course in minimal surfaces[END_REF].) Choose the disc B Σ8 pp 8 q and 1 small enough such that π ´1pB Σ8 pp 8 qq Ă U . We may assume that U " π ´1pB Σ8 pp 8 qq.

From p˚˚q, Σ tn in X U Ă L tn is connected and a graph over B Σ8 pp 8 q, for n large enough. Since BB jt n X U Ă L tn equals Σ tn in X U , it is also connected. Therefore ΦpBB jt n X U q is the set R 2 ˆtx tn u X ΦpU q for some x tn P R. In addition, ΦpΣ 8 X U q equals R 2 ˆtx 8 u X ΦpU q for some x 8 P R. Since lim nÑ8 p tn " p 8 , we have lim nÑ8

x tn " x 8 .

x 8

x t n 1

x tn x 8

x t n 1 x tn ΦpU q ΦpU X B jt n q ΦpU q ΦpU X B jt n q Figure 6.2.

The set U zBB jt n has two components. Therefore, ΦpB jt n X U q is ΦpU q X tx|x 3 ą x tn u or ΦpU q X tx|x 3 ă x tn u. For n large enough, there exists some n 1 ‰ n such that R 2 ˆtx t n 1 u X ΦpU q Ă ΦpB jt n X U q. This implies that B jt n X B jt n 1 is non-empty.

Since S consists of maximal elements in ptB t i u, Ăq, the set B jt n X B jt n 1 is empty which leads to a contradiction. This finishes the proof. 6.3.3. The finiteness of S implies Lemma 6.1.3. We will explain how to deduce Lemma 6.1.3 from the finiteness of S.

Proof. Since S is finite , we may assume that BB j intersects BR k p q transversally for each B j P S. Remark that each B j is equal to some B t i and BB j X BR k p q equals Σ t i X BR k p q. Since each Σ t i is properly embedded, tc i u iPI :" BR k p q X pY B j PS BB j q has finitely many components. Each component is an embedded circle.

The Vanishing property of L and Remark 6.1.2 show that each c i is contractible in BR k p q and bounds a unique closed disc D i Ă BR k p q (since k ě k 0 ). The set pD i , Ăq is a partially ordered set. Let tD j 1 u j 1 PJ 1 be the set of maximal elements. The set J 1 is finite .

Since the boundary of BR k p q X B j is a subset of BB j X BR k p q Ă > iPI c i , it is contained in > j 1 PJ 1 D j 1 .

Next we show that for any B j P S, BR k p q X B j is contained in > j 1 PJ 1 D j 1 .

If not, BR k p qz > j 1 PJ 1 D j 1 is contained in BR k p q X B j for some B j P S. This implies that the composition of two maps π 1 pBR k p qzp> j 1 PJ 1 D j 1 qq Ñ π 1 pB j q Ñ π 1 pR k q is not a zero map. However, the induced map π 1 pB j q Ñ π 1 pR k q is trivial. This is impossible. We conclude that for each B j P S, BR k p q X B j is contained in > j 1 PJ 1 D j 1 .

Therefore, Y B j PS B j X BR k p q is contained in > j 1 PJ 1 D j 1 . From Proposition 6.3.3, L X BR k p q is contained in a disjoint union of finite discs tD j 1 u j 1 PJ 1 . This completes the proof.

Deformation to Positive Scalar curvature

This section follows Kazdan's result [START_REF] Kazdan | Deformation to positive scalar curvature on complete manifolds[END_REF]. In this section, we show that a complete non-Ricci-flat metric of nonnegative scalar curvature can deformed to be a complete metric of positive scalar curvature.

Let pM n , gq be a complete n-manifold. We consider the operator Lpuq " ´∆u `f u where f is a smooth function on M .

For a bounded open set Ω Ă M with smooth boundary and outer normal derivative B{Bν on BΩ, let µ 1 pL, Ωq be the lowest eigenvalue of L with Neumann boundary conditions, Bu{Bν " 0 on BΩ. One has the well-known variational characterization of µ 1 pΩq

µ 1 pL, Ωq " inf v ş Ω p|∇v| 2 `f v 2 qdx ş Ω v 2 dx ,
where dx is the volume form and the infimum is taken over all v in the Sobolev space H 1 pΩq.

Lemma 6.4.1. (See [Theorem A, Page 228] in [START_REF] Kazdan | Deformation to positive scalar curvature on complete manifolds[END_REF]) Assume there is a bounded open set Ω 0 Ă M such that µ 1 pL, Ω 0 q ą 0 and f ě 0 on M zΩ 0 . Then there is a solution u ą 0 on M of Lpuq ą 0; in fact one can find a solution of Lpuq ą 0 satisfying 0 ă C 1 ă u ă C 2 , where C 1 and C 2 are two constants.

In the following, we consider that the conformal Laplacian L g , L g :" ´4pn ´1q n ´2 ∆ g `κg . Theorem 6.4.2. A complete non-Ricci-flat metric of nonnegative scalar curvature can be deformed to be a complete metric of positive scalar curvature.

Proof. Assume that pM, g 0 q is a complete non-Ricci-flat manifold with non-negative (κ g 0 ě 0). Let p be a point in pM, g 0 q satisfying that (6.4.1) Ric g 0 ppq ‰ 0.

Proof. We argue by contradiction. Suppose that pM 3 , gq is a complete contractible genus one 3-manifold of nonnegative scalar curvature.

If g is Ricci-flat, it is a flat metric. Thus, M is homeomorphic to R 3 . This is in contradiction with Theorem 1.3.13(M is not homeomorphic to R 3 ).

If not, g can be deformed to be a complete metric of positive scalar curvature. This contradicts Theorem B 2 .

As the above argument, we have that Corollary 6.4.4. A complete contractible 3-manifold with non-negative scalar curvature and trivial π 8

1 is homeomorphic to R 3 .

6.5. Further questions 6.5.1. The General Case. In [Wan19a, Wan19b], we verified Question 1 for 3manifolds with trivial π 8 1 . The remaining case is contractible 3-manifolds with non-trivial π 8

1 .

In this case, we require more techniques combining minimal surfaces theory and topological surgeries on 3-manifolds.

The key point is to understand stable minimal surfaces (as constructed in Chapter 4.2) and its relationships with the fundamental group at infinity.

Based on this relationship, we attempt to devise a new topological surgery which can reduce Question 1 to Theorem B 2 . 6.5.2. RCD metrics. By [START_REF] Liu | 3-manifolds with nonnegative ricci curvature[END_REF], the Whitehead manifold does not admit any complete metric with positive Ricci curvature. In metric geometry, Riemannian manifolds with lower bounds on Ricci curvature correspond to the RCD spaces. It would be interesting to know whether the Whitehead manifold has a RCD(0, 3) metric. 6.5.3. Spherical Decomposition. Definition 6.5.1. An embedded 2-sphere S in a 3-manifold M is called compressible if S bounds a 3-ball in M .

A spherical decomposition S of a 3-manifold is a locally finite collection of (possibly non-separating) pairwise disjoint embedded 2-sphere in M such that the operation of cutting M along S and gluing a ball to each boundary component of the resulting manifold yields a collection of irreducible manifolds.

A 3 manifold M is irreducible if and only if all 2-spheres in M are compressible. If S is a spherical decomposition, then the collection of sphere obtained by removing compressible spheres in S is still a spherical decomposition.

The prime decomposition theorem for 3-manifold (See [START_REF] Hatcher | Notes on basic 3-manifold topology[END_REF] or [START_REF] Kneser | Geschlossene flächen in dreidimensionalen mannigfaltigkeiten[END_REF]) is equivalent to the statement that every compact 3-manifold has a spherical decomposition. This result does not generalize to open manifolds (See [START_REF] Scott | Fundamental groups of non-compact 3-manifolds[END_REF] and [START_REF] Maillot | A spherical decomposition for riemannian open 3-manifolds[END_REF][START_REF] Maillot | Some open 3-manifolds and 3-orbifolds without locally finite canonical decompositions[END_REF]). The first example was given by Scott [START_REF] Scott | Fundamental groups of non-compact 3-manifolds[END_REF].

A question posed by Maillot is the following:

Question: Does a complete 3-manifold of uniformly positive scalar curvature have a spherical decomposition ?

For a point x P S 1 , f px, ¨q can be considered as an element in MappS 2 , M qq. The projection p 0 : S 1 ˆS2 Ñ S 1 induces the map p 0 : MappS 1 , M q Ñ MappS 1 ˆS2 , M q f Þ Ñ p 0 ˝f Lemma 7.2.1. If π 2 pM q is trivial, the map p 0 is a homotopy equivalence.

Proof. First, we show that M and MappS 2 , M q are homotopic equivalent . For a fixed point x 0 P S 2 , we consider the map

p : MappS 2 , M q ÝÑ M f Þ ÝÑ f px 0 q
For any point x P M , p ´1pxq is a set of all continuous maps from pS 2 , x 0 q to pM, xq. Since π 2 pM q is trivial, the fiber p ´1pxq is contractible for each x P M . Therefore, p satisfies the homotopy lifting property (i.e. p is a fibration). Applying the Puppe sequence (See [Theorem 4.41, Page 376] in [START_REF] Hatcher | Algebraic topology[END_REF]), one has π k pp ´1pxqq Ñ π k pMappS 2 , M qq pÝ Ñ π k pM q Ñ π k´1 pp ´1pxqq Since p ´1pxq is contractible, the map p ˚is an isomorphism for each k. Whitehead's theorem (See [Theorem4.5, Page 346] in [START_REF] Hatcher | Algebraic topology[END_REF]) shows that the map p : MappS 2 , M q Ñ M is a homotopy equivalence.

As a consequence, the induced map p ˚: MappS 1 ˆS2 , M q Ñ MappS 1 , M q is also a homotopy equivalence, since MappS 1 ˆS2 , M q is homeomorphic to MappS 1 , MappS 2 , M qq.

We know that p 0 ˝p˚" Id MappS 1 ,M q . Since p ˚is a homotopy equivalence, then p 0 is also a homotopy equivalence. The lemma follows.

7.2.2. Proof of Theorem F . We begin by a compact manifold pM 4 , gq with b 1 pM q ą 0. There is an embedded circle γ Ă M with the property that rγs is a torsion-free element in H 1 pM q. We use the Poincaré duality to find a class u P H 3 pM 4 q -H 1 pM 4 q satisfying that prγs, uq " 1.

We apply a theorem of Fleming-Federer (See [START_REF] Federer | Normal and integral currents[END_REF]) and the regularity theory for area-minimizing currents (See Chapter 7 of [START_REF] Simon | Lectures on geometric measure theory[END_REF]). This result asserts that, in a Riemannian manifold pX n , gq, for a non-trivial class in H n´1 pX, Zq, there is a volumeminimizing hypersurface in the class satisfying that it is smooth outside a set of Hausdorff dimension ď n ´8 . Therefore, there is a stable minimal hypersurface Σ 3 Ă pM 4 , gq in the class u. In addition, the intersection number pγ, Σq of Σ and γ is `1.

If pM 4 , gq has positive scalar curvature, then Σ admits a metric of positive scalar curvature (See Proposition 3.3.5). In this case, Σ is a connected sum of some spherical manifolds and some copies of S 1 ˆS2 (See [START_REF] Perelman | The entropy formula for ricci flow and its geometric applications[END_REF][START_REF] Perelman | Ricci flow with surgery on three-manifolds[END_REF][START_REF] Perelman | Finite extinction time for the solutions to the ricci flow on certain threemanifolds[END_REF], [START_REF] Tian | Ricci flow and the Poincaré conjecture[END_REF], [BBB `10]).

In the following, we prove Theorem F .

Proof. We argue by contradiction. Suppose a closed aspherical manifold M 4 with b 1 pM 4 q ą 0 has a metric g of positive scalar curvature. 7.3. Further questions 7.3.1. Stable minimal hypersurfaces in R 4 . When using the minimal hypersurfaces to study 4-manifolds, understanding the geometry of minimal hypersurfaces is necessary and crucial. However, the geometry of stable minimal hypersurfaces in a 4-manifold is not known. R. Schoen has conjectured:

Conjecture. (See [Conjecture 2.12, Page 79] of [START_REF] Tobias | A course in minimal surfaces[END_REF]):If Σ 3 Ă R 4 is a complete immersed stable minimal hypersurface with trivial bundle, then Σ is flat. 7.3.2. Closed Aspherical 4-manifolds. Generally, Gromov-Lawson conjecture is still unknown, particularly for 4-manifolds with zero first Betti number.

The proof of Theorem F involves stable minimal hypersurfaces and the geometrization conjecture. In the general case, the main issue is the existence of stable minimal hypersurfaces. In the proof of Theorem F , its existence is ensured by our hypothesis that the first Betti number is non-zero.

In order to overcome it, we attempt to find a covering space whose first betti number is nonzero. We next use topological conditions to construct a complete stable minimal surface. Then one is led to study stable minimal surface in a 4-manifold with uniformly positive scalar curvature.

Combining with the argument in [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF] and some metric inequalities in [START_REF] Gromov | Metric inequalities with scalar curvature[END_REF], we plan to argue by contradiction. We expect to show that such a minimal surface does not exist, which would lead to a contradiction. 7.3.3. Exotic R 4 . An exotic R 4 is a differential manifold that is homeomorphic but not diffeomorphic to the Euclidean space R 4 . The first example were found by Freedman (See [START_REF] Michael | Topology of 4-Manifolds (PMS-39)[END_REF]). Actually, there are infinitely many non-diffeomorphic differential structures of R 4 , as was shown first by Taubes [START_REF] Henry | Gauge theory on asymptotically periodic t4u-manifolds[END_REF].

An interesting question is whether an exotic R 4 admits a complete metric of positive scalar (or Ricci) curvature.

For this question, we might follow the following scheme. First, based on the construction of an exotic R 4 , we attempt to construct a complete stable minimal hypersurface. Then, one is led to understand the geometry of such a hypersurface and its relationship with the differential structures and the positivity of scalar (or Ricci) curvature.

We plan to argue by contradiction. Combining the geometry of the stable minimal hypersurface, we expect to show that such a hypersurface does not exist, which would lead to a contradiction.

Figure

  Figure 0.1.

Figure 0

 0 Figure 0.2.

Figure

  Figure 0.4.

Figure

  Figure 0.5.

Figure 1

 1 Figure 1.2.

Figure

  Figure 1.3.

Figure

  Figure 1.4.

  Lemma 1.3.11. Suppose that the closed solid torus N 1 Ă Int N is homotopically trivial in the closed solid torus N . If IpN 1 , N q ą 0, then any meridian disc D of N contains a meridian of N 1 .It follows fromLemma 1.3.10 and Lemma 1.2.5. 1.3.4. Genus one 3-manifold. (See [GRW18]) Let us describe McMillan's construction in [MJ62].

  H 2.2.1. Surgeries. Consider two closed handlebodies N 1 and N in a 3-manifold M with N 1 Ă Int N . We introduce two types of surgeries on handlebodies: Type I: If there exists a meridian disc D Ă N zN 1 of N , then we consider an open tubular neighborhood N pDq Ă N zN 1 of D. We then have two cases: Case (1): If D is a splitting meridian disc, N zN pDq has two components. The closed handlebody W 1 is defined as the component containing N 1 ; Case(2): If D is a non-splitting meridian disc, N zN pDq is connected. The closed handlebody W 1 is defined by N zN pDq.

  1. Mean Curvature. Let us consider a k-dimensional submanifold Σ k Ă pM n , gq possibly with boundary.

  Theorem 3.1.7. ([Theorem 1, 315] in [Nit69])Let u be a function from D Ă R 2 to R as in Theorem 3.1.6. If Γ is a regular Jordon curve of class C k,α , where k ě 1 and 0

  Lemma 3.2.2. If u 1 and u 2 are two solutions to the minimal surface equation (See Equation (3.1.13)) on a domain Ω Ă R n , then v :" u 1 ´u2 satisfies an equation of the form:(3.2.6) divpApxq∇vq " 0where each eigenvalue λ k of the matrix Apxq :" pa i,j pxqq nˆn satisfies µ ă λ k ă 1{µ , where µ depends only on the upper bounds for the gradient of |∇u 1 | and |∇u 2 |.

  3.1). By Lemma 3.3.1, |u| is smooth and Lp|u|q " ´λ1 |u|. Since |u| ě 0 and |u| is not identically zero, the Harnack inequality (See [Theorem 8.20, Page 199] of [GT15]) implies |u| ą 0 in Ω. Hence, u can not change sign. Next, let us consider the positive solution to the stable operator L (See Equation (3.1.21) or Equation (3.1.22)).

  Theorem 3.3.4. (See [Theorem 1, Page 201] [FCS80]) Let Σ n Ă pM, gq be a complete non-compact minimal hypersurface with trivial normal bundle, then the following are equivalent:

  Theorem 3.3.8. (See [Theorem 10.2, Page 384] of [GL83] or [Theorem 1, Page 228] of [Ros06]) Let pM 3 , gq be a Riemannian 3-manifold with the scalar curvature κ ě c where c is a positive constant. If Σ is a stable minimal surface immersed in M , then for x P Σ: d Σ px, BΣq ď 2π ? 3c .

B

  Figure 6.1.

variétés. Ce fait est apparu dans les articles de Schoen et Yau [SY82, SY79b, SY79a] ainsi que de Gromov et Lawson cite GL et divers autres travaux.

  , nous montrons que le nombre de composantes connexes de Ω k X N 1 intersectant N 0 tend vers l'infini quand k tend vers l'infini (voir Chapitre 2.1). Par conséquent, il existe une infinité de composantes connexes de Σ X N 1 intersectant N 0 . Par un résultat de Meeks et Yau (voir Théorème 3.2.7 et[START_REF] Meeks | Topology of three dimensional manifolds and the embedding problems in minimal surface theory[END_REF]), chacune de ces composantes contient une certaine quantité d'aire. Ainsi, Σ X N 1 a une aire infinie. D'autre part, puisque pM, gq a courbure scalaire positive , non seulement Σ est conformément difféomorphe à R 2 (voir Corollaire 3.3.11), mais sa géométrie est également contrainte par l'inégalité extrinsèque de Cohn-Vossen: , cela est en contradiction avec la surface infinie contenue dans Σ X N 1 .Si BN k n'est pas mean convex, on modifies la métrique dans un plus petit voisinage tubulaire de BN k afin que, pour la nouvelle métrique, elle devienne mean convex.. Alors Ω k est minimal stable pour la nouvelle métrique, et pour la métrique originale il l'est loin du voisinage BN k (par exemple, près de N k ), ce qui est suffisant pour notre preuve.

	Whitehead n'est pas simplement connexe l'infini mais son groupe fondamental à l'infini est trivial. Nous prouvons le résultat suivant : Théroème 4 (=Theorem C) Une 3-variété contractile à courbure scalaire positive et π 8 à 1 trivial est homéomorphe à R 3 . Ce résultat peut également être généralisé à la courbure scalaire non négative. Corollarie (voir Corollaire 6.4.4)Une 3-variété contractile à courbure scalaire non négative et π 8 1 trivial est homéomorphe à R 3 . Cependant, il existe une quantité indénombrable de 3-variétés contractiles deux à deux non homéomorphes dont le π 8 1 n'est pas trivial. Dans le Chapitre 1.3, nous construisons une telle variété et montrons que cette variété n'a pas de métrique complète de courbure scalaire positive (voir Théorème 1.3.15). 0.7. L'idée de la preuve du Théorème 3 Il est classique que la théorie des surfaces minimales donne des informations topologiques sur les 3-Pour la preuve du théorème 3, raisonnons par l'absurde. Supposons que pM, gq soit une 3-variété riemannienne complète à courbure scalaire positive, où M :" Ť k N k est une 3-variété contractile de genre un et la famille tN k u k de tores solides est supposée comme dans Théorème 1.3.13. 0.7.1. Surfaces minimales et limites. Comme dans [SY82] et [GL83], notre première étape consiste à construire des surfaces minimales. Choisissons γ k Ă BN k comme méridien de N k (voir Définition 1.2.1). En gros, il borde un disque minimal stable plongé Ω k . Son existence est assurée par le résultat de Meeks et Yau (voir [MIY80, MIY82] ou le théorème 3.1.8) lorsque la frontière BN k est mean convex. Considérons le cas le plus simple où Ω k converge vers une surface minimale stable connexe Σ. (non compacte). Si la variété complète pM 3 , gq a courbure scalaire positive ou nulle (κpxq ě 0), alors ż Σ κpxq `1{2|A| 2 dv ď 2πχpΣq. où |A| 2 est le carré de la norme de la seconde forme fondamentale de Σ. De plus, si κ ą 0 et Σ est plongée, alors Σ est un plan proprement plongé. Puisque la courbure scalaire est minorée par une constante strictement positive sur N 1 0.7.2. Propriétés de la surface limite. En général, Ω k sous-converge vers une lamination minimale L :" Ť t inΓ L t (c'est-à-dire une union disjointe de certaines surfaces minimales plongées). Elle peut avoir une infinité de composantes. Cependant, chaque feuille L t de L est une surface minimale stable complète (non compacte) (voir Théorème 4.2.3). Puisque pM, gq a une métrique à courbure scalaire positive, elle est homéomorphe à R 3 (voir Corollaire 3.3.11). La géométrie de chaque feuille est influencée par l'inégalité extrinsèque de Cohn-Vossen (cf. Théorème 5) ainsi que par une propriété topologique de M , appelée Propriété P (voir D'une partThéroème 5 (=Theorem D) Soit Σ 2 Ă pM 3 , gq une surface minimale complète immergée Définition 2.1.3).

Ces deux aspects nous indiquent que la lamination L a la propriété d'annulation pour tN

  k u k . C'est-à-dire, il existe un entier positif k 0 tel que pour tout k ě k 0 et tout t P Γ, tout cercle dans L t X BN k est homotopiquement trivial à BN k .

  Expliquons comment déduire une contradiction avec la propriété d'annulation. Nous montrons que pour tout k ě k 0 , Ω k X BN k 0 contient une courbe ferme qui n'est pas homotopiquement triviale dans BN k 0 (voir le Lemme 1.3.11). Grosso modo, ces cercles non homotopiquement triviaux vont sous-converger vers une courbe fermée dans L XBN k 0 qui n'est pas contractile dans BN k 0 . Il s'ensuit que Ω k sous-converge vers la lamination L . Par conséquent, certaines feuilles de L ont un cercle non contractile dans BN k 0 . Ceci est en contradiction avec la propriété d'annulation ci-dessus de L .

	0.8. π 8 1 et la propriété d'annulation
	0.8.1. Corps à anses et propriété H. Soit pM, gq une 3-variété complète contrac-
	tile à courbure scalaire positive. Il s'agit d'une union croissante de corps à anses tN k u
	(voir Théorème 1.1.12).
	Dans ce qui suit, nous considérons que M n'est pas homéomorphe à R 3 . Nous pouvons
	supposer qu'aucun des N k n'est contenu dans une 3-boule (c'est-à-dire homéomorphe	à
	une boule unitaire dans R 3 ) dans M (voir Remarque 1.1.8). Cela joue un rôle crucial dans
	notre argument.

  Dans le cas du genre un, la géométrie d'une surface minimale stable est contrainte par l'indice géométrique (voir Propriété P dans[START_REF] Wang | Contractible 3-manifolds and positive scalar curvatures (I)[END_REF] ou la Définition 2.1.3). Dans le cas du genre supérieur, le comportement d'une surface minimale stable est lié au groupe fondamental à l'infini.Afin de clarifier leur relation, introduisons une propriété géométrique, appelée la propriété d'annulation. Premièrement, nous considérons une 3-variété complte contractile pM, gq qui n'est pas homéomorphe à R 3 . Comme indiqué ci-dessus, il existe une famille croissante tR k u k de corps à anses avec la propriété H.On dit qu'une surface minimale stable complète plongée Σ Ă pM, gq satisfait la propriété d'annulation pour la famille tR k u k s'il existe un entier positif kpΣq tel que pour k ě kpΣq, tout cercle dans ΣXBR k soit homotopiquement trivial dans BR k (

	0.8.2. La proriété d'annulation. Voir Chapitre
	5).
	Si une surface minimale stable complète ne satisfait pas la propriété d'annulation
	pour tR k u k , elle donne un élément non trivial dans π 8 1 pM q (voir le Lemme 5.2.1). En
	conséquence, si π 8 1 est trivial, toute surface minimale stable complète dans M a la pro-
	priété d'annulation pour tR k u k (voir le Corollaire 5.2.2).
	0.8.3. L'idée de la preuve du théorème 4. Raisonnons par l'absurde. Supposons
	qu'une 3-variété pM, gq complète contractile à courbure scalaire positive et π 8 1 pM q trivial, ne soit pas homéomorphe à R 3 .
	Avant de construire des surfaces minimales, introduisons une notation de topologie en
	dimension 3.
	Un système de méridiens d'un corps à anses N est une collection de g méridiens
	distincts tγ l u g l"1 tels que BN z > g l"1 γ l soit hoéomorphe à un disque ouvert privé de certains
	sous-disques fermés (voir le Lemme 1.2.7). Son existence est assurée par le Lemme 1.2.7.
	Soit tN k u k et tR k u k comme ci-dessus. Puisque N 0 n'est pas contenu dans une 3-boule
	(voir la remarque 1.1.8), le genre de N k est suprieur zro. Le corps à anses N k a un système
	de méridiens tγ l k u BΩ l k " γ l k . Leur existence est assurée par les travaux de Meeks et Yau [MIY80, MIY82] gpN k q l"1 . En gros, il existe gpN k q disques disjoints d'aire minimale tΩ l k u l avec
	(voir le théorème 6.28 de [CM11]) lorsque la frontière BN k est mean convex.
	Expliquons leur existence. Nous construisons ces disques par récurrence sur l.
	Lorsque l " 1, il existe un disque plongé d'aire minimale Ω 1 k Ă N k avec BΩ 1 k " γ 1 k Si M n'est pas homéomorphe à R 3 , l'existence d'une telle famille est assure (voir [MIY80, MIY82] ou le théorème 6.28 de [CM11]).
	par le Thorme 2.2.6. Elle n'est pas unique. De plus, l'union d'une telle famille peut ne pas correspondre à M . Supposons qu'il existe l disques stables plongés disjoints d'aire minimale tΩ i k u l i"1 avec BΩ i k " γ i k . Notre objectif est de construire une surface minimale stable Ω l`1 k
	Par exemple, si M :"	Ť k N k est une 3-variété contractile de genre un, la famille tN k u
	(supposée comme dans le Théorème 1.3.13) vérifie la propriété ci-dessus (propriété H)
	(voir le lemme 2.10 dans [Wan19a] ou Lemme 1.3.10).

construction par récurrence. Comme dans le cas du genre un, si BN k n'est pas mean convex, nous pouvons dformer la mtrique dans un petit voisinage de celle-ci pour qu'elle devienne mean convex.

  il existe un entier positif k 0 tel que pour tout k ě k 0 et tout t P Λ, n'importe quel cercle dans L t X BR k soit homotopiquement trivial dans BR k .Raisonnons par l'absurde. Supposons qu'il existe une suite croissante tk n u n d'entiers et une suite tL tn u de feuilles dans L telles que L tn X BR kn ait au moins un cercle non homotopiquement trivial dans BR kn pour chaque n.

			proviennent tous du même disque
	minimal Ω i k . Par conséquent, la courbure moyenne de la frontière de pT k,l , g| T k,l q est
	positive. (Voir Chapitre 4.2)
	De plus, tγ i k u iąl est un système de méridiens du corps à anses pT k,l , g| T k,l q. Ensuite,
	nous utilisons le résultat de Meeks et Yau pour trouver une surface minimale stable
	plongée Ω l`1 k	Ă T k,l dont le bord est γ l`1 k . Ces disques tΩ i k u l`1 i"1 sont disjoints dans N k .
	Ceci termine la Définissons la lamination L k :"	š l Ω l k (qui est une union disjointe de surfaces plongées).
	Nous montrons que chaque lamination L k intersecte l'ensemble compact R 0 (Corollaire
	1.2.8). Selon la théorie de Colding-Mincozzi (voir l'annexe B de [CM04]), la suite tL k u k
	sous-converge vers une laminiation L :"	Ť tPΛ L t in pM, gq (voir le Théorème 4.2.3).
	Notons que chaque feuille L t est une surface minimale stable complète (non compacte).
	Comme indiqué ci-dessus, puisque pM, gq est a 3-variété complète à courbure scalaire
	positive et que π 8 1 pM q est trivial, chaque feuille L t dans L a la propriété d'annulation
	pour tR k u k (voir le Lemme 5.2.1 et le Corollaire 5.2.2). En outre, la lamination L satisfait
	la propriété d'annulation (voir le Corollaire 5.2.4). C'est-à-dire,
	La raison est dcrite comme suit.

  1 BpΣXBpx 0 ,tqq ă ∇ Σ r 2 , n ą, since |∇ Σ r| ď |∇r| ď 1

	(3.2.14)
				ż
				"	∆ Σ r 2 ,	doing integration by parts,
				ΣXBpx 0 ,tq
				ż
				ě 2nK		r cotpKrq	from Equation(3.2.13).
				Bpx 0 ,tqXΣ
	where n is the outward unit normal vector of BpΣ X Bpx 0 , tqq in Σ.
	Let Cptq "	ş Bpx 0 ,tqXΣ r cotpKrq. Then by the coarea formula (See Equation (3.2.10))
	and the fact that |∇r| ď 1, we have that
	(3.2.15)	BCptq Bt	"	ż tx | rpxq"tuXΣ	r cotpKrq |∇r|	ě t cotpKtqVolpBpΣ X Bpx 0 , tqqq.
	Inserting Equation (3.2.14) into Equation (3.2.15), we find
	(3.2.16)			BCptq Bt		ě nK cotpKtqCptq.
	It is easy to verify that
				lim
	(3.2.13)
				Hesspe i , e i qprq,	since Hpxq " 0
		ě 2nKr cotpKrq,

where x P Σ, ∇ Σ , ∇ N and Hpxq are defined as in Section 3.1, (See also the argument [Page 243] of [SY77]). Integrating Equation (3.2.13) over Bpx, tq and noting that |∇r| ď 1, we have, 2tVolpBpΣ X Bpx 0 , tqq ě ż tÑ0 Cptq sinpKtq ´1 " K ´n´1 C n where C n is a positive constant depending only on n. It follows from Equation (3.2.16) that (3.2.17) Cptq ě K ´n´1 C n psinpKtqq n for all t ď mint , ipM qu. Therefore, Equation (3.2.14) shows that (3.2.18) VolpBpΣ X Bpx 0 , tqqq ě nC n K ´nt ´1psinpK tqq n . By the coarea formula (See Equation (3.2.14)) again,

ET LA PROPRI ÉT É D'ANNULATION

ET LA PROPRI ÉT É D'ANNULATION

The author is supported by ERC Advanced Grant 320939, GETOM.

We have that g 1 ii p0q " gp∇ Ft F x i , F x i q " gp∇ Fx i F t , F x i q " ´gp∇ Fx i F x i , F t q. Since Σ is minimal and F T t " 0, we have ř i g 1 ii p0q " ´gp ř i ∇ Fx i F x i , F t q " ´gpF t , Hq " 0 as in Equation (3.1.6). Therefore, we get (3.1.18) 2 d 2 dt 2 νptq ˇˇˇt "0

Lemma 3.1.3. At the point x, we get

`2 ÿ i gpR M pF x i , F t qF t , F x i q `2div Σ pF tt q.

Proof. An easy computation gives that g 1 ij p0q " gp∇ Ft F x i , F x j q `gpF x i , ∇ Ft F x j q " ´2gpApF x i , F x j q, F t q. This implies the first equation.

We compute that

Next use the definition of the Riemann curvature tensor R M of M to get ÿ i gp∇ Ft ∇ Ft F x i , F x i q " ÿ i gp∇ Ft ∇ Fx i F t , F x i q sincerF t , F x i s " 0 " ÿ i gpRpF x i , F t qF t , F x i q `ÿ i gp∇ Fx i ∇ Ft F t , F x i q " ÿ i gpRpF x i , F Ft qF t , F x i q `div Σ pF tt q.

Therefore, we have

`2 ÿ i gpR M pF x i , F t qF t , F x i q `2div Σ pF tt q.

The map gpAp¨, ¨q, F t q : T x Σ ˆTx Σ Ñ R is a symmetric bilinear map. Since tF x i u is an orthonormal basis of T Σ at x, the squared norm |gpAp¨, ¨q, F t q| 2 pxq is equal to ř ij gpApF x i , F x j q, F t q 2 . Similarly, the trace Tr Σ gpR M p¨, F t qF t , ¨qpxq equals

d 2 dt 2 VolpF pΣ, tqq ˇˇˇt "0 " ´żΣ gpF t , LpF t qq ě 0.

A complete (possibly non-compact) minimal submanifold without boundary is said to be stable if all compact subdomains are stable.

For a minimal hypersurface Σ n´1 Ă pM n , gq with trivial normal bundle, the stability operator simplifies significantly since, in this case, it becomes an operator on functions. Namely, if we identify a normal vector field X " ηN , then

where Ric M is the Ricci tensor of M . Schoen and Yau [START_REF] Schoen | On the structure of manifolds with positive scalar curvature[END_REF] pointed out that the stability operator (See Equation (3.1.21)) is linked with the scalar curvature of M (See Page 7 in [START_REF] Schoen | On the structure of manifolds with positive scalar curvature[END_REF]).

Proposition 3.1.5. (See Page 7-8 of [START_REF] Schoen | On the structure of manifolds with positive scalar curvature[END_REF])Let Σ n Ă pM n`1 , gq be a minimal surface with trivial normal bundle. Then the operator can be written as

where κ Σ and κ M are the scalar curvature of Σ and M , respectively.

Proof. Fixed a point x P Σ, we may choose an orthonormal basis tE i u n`1 i"1 of T x M . The unit vector E n`1 is equal to the unit normal vector of Σ. The Gauss Equations (See Equation (3.1.4)) assert

where K Σ and K M are the sectional curvature of Σ and M respectively, A ij :" ApE i , E j q. Summing Equation (3.1.23), we have

Therefore, by the minimality of Σ, the scalar curvature of M is

(3.1.24)

Therefore, putting Equation (3.1.24) into Equation (3.1.21), we get

From the proof of Theorem 5.1.1, we have that if pM, gq has positive scalar curvature, any complete stable minimal lamination in pM, gq has the Vanishing Property for tR k u.

The vanishing Property and π 8

1 Generally, the geometry of a stable minimal surface is constrained by some topological properties of the 3-manifold. For example, the fundamental group at infinity. In the following, we will study a complete (non-compact) stable minimal surface Σ Ă pM, gq and its relationship with the fundamental group at infinity. Lemma 5.2.1. Let pM, gq be a complete contractible Riemannian 3-manifold with positive scalar curvature κpxq ą 0 and tR k u k a family of handlebodies with Property H. If a complete embedded stable minimal surface Σ does not satisfy the Vanishing Property for tR k u k , then π 8 1 pM q is non-trivial. Roughly, there is a sequence of non-trivial circles in Σ going to infinity. This sequence gives a non-trivial element in π 8 1 pM q. Proof. Since Σ does not satisfy the Vanishing property for tR k u, there exists a sequence tk n u n of increasing integers so that for each k n , there is a circle γ

However, γ n may not be a meridian. We will choose a meridian in D n of R kn to replace it.

Since the map π 1 pBR kn q Ñ π 1 pM zR kn q is injective (See Definition 2.2.5), we use Corollary 1.2.6 to see that D n contains at least one meridian of R kn . Without loss of generality, we may assume that γ n is a meridian of R kn and Int D n has no meridian of R kn . (If not, we can replace γ n by the meridian in Int D n ).

Since tγ n u n is a collection of disjointly embedded circles in Σ, one of the following holds: for each n 1 and n 1 ,

) and the map π 1 pBR kn q Ñ π 1 pM zR kn q is injective, we use Corollary 1.2.6 to see that D n 1 Ă Int D n contains at least one meridian of R kn . This is in contradiction with the above assumption.

We will show there is an increasing subsequence of tD n u. Furthermore, the boundaries of these discs in the subsequence gives a non-trivial element in π 8 1 pM q.

Step 1: the existence of the ascending subsequence of tD n u.

We argue by contradiction. Suppose these is no ascending subsequence in tD n u. Consider the partially ordered set ptD n u n , Ăq induced by the inclusion. Let C be the set of minimal elements in ptD n u n , Ăq. These discs in C are disjoint in Σ.

93

If the set C is finite, we consider the integer n 0 :" maxtn| D n P Cu. From the above fact p˚q, the subsequence tD n u nąn 0 is an increasing subsequence, which contradicts our hypothesis. Therefore, we can conclude that the set C is infinite. That is to say, there is a subsequence tD ns u s of disjointly embedded discs.

From Remark 2.2.7, the map π 1 pBR kn s q Ñ π 1 pM zR 0 q is injective. Therefore, the disc D ns intersects N 0 . Since N 0 is a subset of R 0 , it also intersects R 0 .

Choose x ns P R 0 X D ns and r 0 " 1 2 minti 0 , ru, where r :" d M pBR 0 , BR 1 q and i 0 :" inf xPR 1 pInj M pxqq. Hence, the geodesic ball Bpx ns , r 0 q in M lies in R 1 .

Define the constants C :" inf xPR 1 κpxq, K :" sup xPR 1 |K M | where K M is the sectional curvature of pM, gq.We apply Theorem 3.2.7 (See [Lemma 1, Page 445] of [START_REF] Meeks | Topology of three dimensional manifolds and the embedding problems in minimal surface theory[END_REF]) to the minimal surface D ns X R 1 in pR 1 , BR 1 q and obtain that AreapD ns X Bpx ns , r 0 qq ě C 1 pK, i 0 , r 0 q. This leads to a contradiction from Theorem 3.3.10 as follows:

Therefore, we can conclude that there is an ascending subsequence of tD n u n . From now on, we abuse the notation and write tD n u for an ascending subsequence.

Step 2: π 8 1 pM q is non-trivial. Claim: There is an integer N so that for n ě N , pD n zD n´1 q X R 0 is empty. We argue by contradiction. Suppose that there exists a family tn l u of increasing integers such that D n l zD n l´1 intersects R 0 .

Choose x l P D n l zD n l´1 X R 0 . Hence the geodesic ball Bpx l , r 0 q in M is contained in R 1 , where r 0 is assumed as above. We apply Theorem 3.2.7 (See [Lemma 1, Page 445] in [START_REF] Meeks | Topology of three dimensional manifolds and the embedding problems in minimal surface theory[END_REF]) to the minimal surface D n l zD n l´1 X R 1 in pR 1 , BR 1 q.

AreappD n l zD n l´1 q X Bpx l , r 0 qq ě C 1 pK, i 0 , r 0 q. From Theorem 3.3.10, one gets a contradiction as follows:

This proves Claim 1.

Therefore, for n ą N , γ n is homotopic to γ N in M zR 0 and not nullhomotopic in M zR 0 .

sequence to N " B 0 Y Ω B 1 , one has:

Since Ω is connected, Ĥ0 pΩq is trivial. Therefore, H 1 pN q is also trivial. This contradicts the fact that H 1 pN q is isomorphic to Z gpN q . This completes the proof.

Consider two disjoint surfaces pΩ 1 , BΩ 1 q and pΩ 2 , BΩ 2 q as assumed in Lemma 6.2.1. For t " 1, 2, N zΩ t has two components. Let B t be the unique component of N zΩ t satisfying π 1 pB t q Ñ π 1 pN q is trivial. One has: Lemma 6.2.2. Let pΩ 1 , BΩ 1 q and pΩ 2 , BΩ 2 q be two disjoint surfaces as assumed in Lemma 6.2.1. For each t " 1, 2, N zΩ t has a unique component B t with the property that the map π 1 pB t q Ñ π 1 pN q is trivial. Then it holds one of the following:

(

Proof. Suppose B 1 X B 2 and B 1 zB 2 are both nonempty. Say, there are two points

First, Ω 2 is contained in B 1 . The reason follows as below: B 1 includes a curve γ joining p 1 and p 2 (since B 1 is connected). γ must intersect Ω 2 at some point(s). Hence, Ω 2 X B 1 is not empty. Since Ω 1 X Ω 2 is empty, Ω 2 lies in one of component of N zΩ 1 . Therefore, Ω 2 is contained in B 1 .

Second, Ω 2 cuts B 1 into two components. Otherwise, there is a circle in B 1 which intersects Ω 2 at one point. As argued in Lemma 6.2.1, such a circle can not exist.

Finally, take the component B of B 1 zΩ 2 satisfying that BB X Ω 1 is empty. Then, B is also a component of N zΩ 2 . In addition, the map π 1 pBq Ñ π 1 pB 1 q Ñ π 1 pN q is trivial. From the uniqueness of B 2 , one has that B " B 2 . This implies B 2 Ă B 1 .

Proof of Lemma 6.1.3

In order to prove Lemma 6.1.3, we will introduce the set S and prove the finiteness of S. The finiteness of S will imply Lemma 6.1.3. 6.3.1. Definition of the set S. Let pM, gq, L , tL k u and tN k u be assumed as in the proof of Theorem D. As in Chapter 6.2, there is an ascending family tR k u k of handlebodies with Property H with the property that L satisfies the Vanishing property for tR k u k . That is to say, there is a positive integer k 0 so that for each k ě k 0 and each t P Λ, each circle in L t XBR k is contractible in BR k .

In the following, we will work on the open handlebody Int R k and construct the set S, for a fixed integer k ě k 0 .

Choose the geodesic ball Bpp, rq centered at p with radius r, where r is less than the injective radius of pM, gq at p. Since κ g 0 ě 0, then µ 1 pL g 0 , Bpp, rqq is non-negative.

If µ 1 pL g 0 , Bpp, rqq ą 0, we use Lemma 6.4.1 to find a positive function u with the following properties: ' L g 0 puq ą 0; ' there is a positive constant C such that u ě C. Choose the new metric g :" u 4 n´2 g 0 . It is a complete metric over M . The scalar curvature is κ g " L g 0 puqu ´n`2 n´2 ą 0. The metric g is the required candidate as in the statement.

If µ 1 pL g 0 , Bpp, rqq " 0, we have that the scalar curvature κ g 0 " 0 on Bpp, rq and the eigenfunction φ for µ 1 pL g 0 , Bpp, rqq is a constant function. The reason is as follows:

We have that $ & % ´∆g 0 φ `κg 0 φ " 0

Bφ{Bν " 0 Dong integration by parts, we have that ş Bpp,rq |∇φ| 2 `κg 0 φ 2 " 0. Thus, φ is a constant function and κ g 0 " 0 on Bpp, rq.

From [Lemma 3.3, Page 232] in [START_REF] Kazdan | Deformation to positive scalar curvature on complete manifolds[END_REF], there exists a new metric g t 0 so that (1) µ 1 pL gt 0 , Bpp, rqq ą 0;

(2) g t 0 is equal to g 0 outside Bpp, rq. As the above case, we could find a complete metric with positive scalar curvature.

The metric g t 0 is constructed as follows: Pick a a function η P C 8 0 pBpp, rq, R ě0 q with ηppq ą 0 and consider a family of metrics g t " g 0 ´t ¨η ¨Ric g 0 Since κ g 0 " 0 on Bpx, rq, the first variation formal (See [Page 233] of [START_REF] Kazdan | Deformation to positive scalar curvature on complete manifolds[END_REF] or [START_REF] Kazdan | Prescribing curvatures[END_REF]) gives that (6.4.2)

Since ηppq ą 0, Equations (6.4.1) and (6.4.2) gives d dt µ 1 pL gt , Bpp, rqq| t"0 ą 0.

Since µ 1 pL g 0 , Bpp, rqq " 0, we find that µ 1 pL gt 0 , Bpx, rqq ą 0 for some t 0 ą 0. The metric g t 0 is the required metric.

As a consequence, we have that Corollary 6.4.3. No contractible genus one 3-manifold admits a complete metric of nonnegative scalar curvature.

For this question, a possible approach is to use the minimal surface theory to study the geometry of incompressible spheres.

This question is related to the open problem, introduced by Bessières, Besson and Maillot [START_REF] Bessières | Ricci flow on open 3-manifolds and positive scalar curvature[END_REF]:

Question :(See [START_REF] Bessières | Ricci flow on open 3-manifolds and positive scalar curvature[END_REF]) Let M be a connected, orientable 3-manifold which admits a complete metric of uniformly positive scalar curvature. Is M a connected sum of spherical manifolds and copies of S 2 ˆS1 ? Generally, we attempt to use the spherical decomposition to study the following question:

Question: (See Problem 27 in [START_REF] Shing | Problem section[END_REF]) Classify 3-manifolds admitting complete Riemannian metrics of positive (resp. nonnegative) scalar curvature up to diffeomorphism.

Part 4

Closed aspherical 4-Manifolds CHAPTER 7

Gromov-Lawson Conjecture 7.1. Aspherical 4-manifolds Definition 7.1.1. A space M is called aspherical if it is path-connected and all its higher homotopy groups vanish (i.e. π k pM q is trivial for k ě 2). For example, a CW complex M is aspherical if and only if its universal cover is contractible.

Lemma 7.1.2. If M n is an aspherical manifold, then its fundamental group is torsionfree.

However, if M n is an aspherical manifold, the first homology group H 1 pM q may be not torsion-free. For example, the Klein bottle K 2 is aspherical (since its universal cover is R 2 ). However, H 1 pK 2 q -Z ' Z{2Z. For example, a n-dimensional torus T n has no metric of positive scalar curvature for n ě 2. It was proved by Gromov and Lawson [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF] and by Schoen and Yau [START_REF] Schoen | Complete three-dimensional manifolds with positive ricci curvature and scalar curvature[END_REF][START_REF] Schoen | Positive scalar curvature and minimal hypersurface singularities[END_REF].

It is well-known from [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF] that this conjecture holds for the 3-dimensional case. In dimension four, Gromov and Lawson [START_REF] Gromov | Positive scalar curvature and the dirac operator on complete riemannian manifolds[END_REF] gave a partial solution requiring an additional hypothesis about incompressible surfaces.

We prove that Theorem. F No closed aspherical 4-manifolds with nonzero first Betti number has a metric of positive scalar curvature.

Remark that there is some closed aspherical 4-manifold whose first homology group vanishes (See [RT05]).

7.2. The non-existence result 7.2.1. Topological preliminary. For two topological spaces X and Y , let us consider the topological space MappX, Y q, consisting of all continuous maps from X to Y and the compact-open topology.

For a manifold M , the space MappS 1 ˆS2 , M q is homeomorphic to the space MappS 1 , MappS 2 , M qq. The homeomorphism is given by

Let γ and Σ be constructed as above. The intersection number pΣ, γq of Σ and γ is not zero. In addition, Σ has a metric of positive scalar curvature. By [Per02a, Per02b, Per03], we see that

where each Γ i is a finite subgroup of Op4q for 1 ď i ď j.

There exists a family of disjointly embedded splitting 2-spheres tS i u n´1 i"1 in Σ. They cut M into n-components, denoted by tX i u

k is not a subset of M 4 .) Since π 2 pM q is trivial, each 2-sphere S k bounds an immersed 3-ball B k Ă M 4 . Let us consider a submanifold

Each X 2 k Ă M 4 can be viewed as the image of some map f k from X 1 k to M 4 . In addition, we know that rΣs " ř n k"1 rX 2 k s in H 3 pM 4 , Zq. Since the intersection number pγ, Σq is not equal to zero, there is some k such that the intersect number pγ, X 2 k q is not zero. As mentioned above, X 1 k is a spherical 3-manifold or S 1 ˆS2 .

Case I: If X 1 k is a spherical manifold (that is, it is S 3 {Γ k ), X 2 k can be considered as the image of the map from S 3 to M . However, since π 3 pM 4 q is trivial, we see that X 2 k is contractible in M 4 . Therefore, the intersection number pγ, X 2 k q is zero. This is in contradiction with the last paragraph.

Case II: If X 1 k is a S 1 ˆS2 , then f k is an element in MappS 1 ˆS2 , M 4 q. Since π 2 pM 4 q is trivial, Lemma 7.2.1 shows that there is a map f 1 k P MappS 1 , M 4 q such that p 0 pf 1 k q is homotopic to f k in M 4 .

We may assume that the image of p 0 pf 1 k q is an embedded circle γ 1 . Therefore, X 2 k is homotopic to γ 1 in M 4 . Hence, the intersection number pγ, X 2 k q is equal to the intersection number pγ, γ 1 q. Since dimpM 4 q " 4, we see that the intersection number pγ, γ 1 q is zero in M 4 . This contradicts the above fact that pγ, X 2 k q ‰ 0. This finishes the proof of Theorem F .