Laurent Bessières

Gérard Besson

Sylvain Maillot

Ricci flow on open 3-manifolds and positive scalar curvature

We show that an orientable 3-dimensional manifold M admits a complete riemannian metric of bounded geometry and uniformly positive scalar curvature if and only if there exists a finite collection F of spherical space-forms such that M is a (possibly infinite) connected sum where each summand is diffeomorphic to S 2 ×S 1 or to some member of F. This result generalises G. Perelman's classification theorem for compact 3-manifolds of positive scalar curvature. The main tool is a variant of Perelman's surgery construction for Ricci flow.

Introduction

Thanks to G. Perelman's proof [START_REF] Perelman | The entropy formula for the 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 three-manifolds[END_REF] of W. Thurston's Geometrisation Conjecture, the topological structure of compact 3-manifolds is now well understood in terms of the canonical geometric decomposition. The first step of this decomposition, which goes back to H. Kneser [START_REF] Kneser | Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten[END_REF], consists in splitting such a manifold as a connected sum of prime 3-manifolds, i.e. 3-manifolds which are not nontrivial connected sums themselves.

It has been known since early work of J. H. C. Whitehead [Whi35] that the topology of open 3-manifolds is much more complicated. Directly relevant to the present paper are counterexamples of P. Scott [START_REF] Scott | Some examples of exotic noncompact 3-manifolds[END_REF] and the third author [START_REF] Maillot | Some open 3-manifolds and 3-orbifolds without locally finite canonical decompositions[END_REF] which show that Kneser's theorem fails to generalise to open manifolds, even if one allows infinite connected sums.

Of course, we need to explain what we mean by a possibly infinite connected sum. If X is a class of 3-manifolds, we will say that a 3-manifold M is a connected sum of members of X if there exists a locally finite graph G and a map v → X v which associates to each vertex of G a copy of some manifold in X , such that by removing from each X v as many 3-balls as vertices incident to v and gluing the thus punctured X v 's to each other along the edges of G, one obtains a 3-manifold diffeomorphic to M. This is equivalent to the requirement that M should contain a locally finite collection of pairwise disjoint embedded 2-spheres S such that the operation of cutting M along S and capping-off 3-balls yields a disjoint union of 3-manifolds which are diffeomorphic to members of X . 1Note that restricting this definition to finite graphs and compact manifolds yields a slightly nonstandard definition of a connected sum. In the usual definition of a finite connected sum, one has a tree rather than a graph. It is well-known, however, that the graph of a finite connected sum (in the sense of the previous paragraph) can be made into a tree at the expense of adding extra S 2 × S 1 factors. The more general definition we have chosen for this paper seems more natural in view of the surgery theory for Ricci flow. It can be shown that the two definitions are equivalent even when the graph is infinite; however, having a tree rather than a graph is only important for issues of uniqueness, which will not be tackled here.

The above-quoted articles [START_REF] Scott | Some examples of exotic noncompact 3-manifolds[END_REF][START_REF] Maillot | Some open 3-manifolds and 3-orbifolds without locally finite canonical decompositions[END_REF] provide examples of badly behaved open 3-manifolds, which are not connected sums of prime 3-manifolds. From the point of view of Riemannian geometry, it is natural to look for sufficient conditions for a riemannian metric on an open 3-manifold M that rule out such exotic behaviour. One such condition was given by the third author in the paper [START_REF] Maillot | A spherical decomposition for Riemannian open 3-manifolds[END_REF]. Here we shall consider riemannian manifolds of positive scalar curvature. This class of manifolds has been extensively studied since the seminal work of A. Lichnerowicz, M. Gromov, B. Lawson, R. Schoen, S.-T. Yau and others (see e.g. the survey articles [START_REF] Gromov | Sign and geometric meaning of curvature[END_REF][START_REF] Rosenberg | Manifolds of positive scalar curvature: a progress report[END_REF].)

Let (M, g) be a riemannian manifold. We denote by R min (g) the infimum of the scalar curvature of g. We say that g has uniformly positive scalar curvature if R min (g) > 0. Of course, if M is compact, then this amounts to insisting that g should have positive scalar curvature at each point of M.

A 3-manifold is spherical if it admits a metric of positive constant sectional curvature. M. Gromov and B. Lawson [START_REF] Gromov | Spin and scalar curvature in the presence of a fundamental group[END_REF] have shown that any compact, orientable 3-manifold which is a connected sum of spherical manifolds and copies of S 2 × S 1 carries a metric of positive scalar curvature. Perelman [START_REF] Perelman | Ricci flow with surgery on three-manifolds[END_REF], completing pioneering work of Schoen-Yau [START_REF] Schoen | Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature[END_REF] and Gromov-Lawson [START_REF] Gromov | Positive scalar curvature and the Dirac operator on complete Riemannian manifolds[END_REF], proved the converse.

In this paper, we are mostly interested in the noncompact case. We say that a riemannian metric g on M has bounded geometry if it has bounded sectional curvature and injectivity radius bounded away from zero. It follows from the Gromov-Lawson construction that if M is a (possibly infinite) connected sum of spherical manifolds and copies of S2 × S 1 such that there are finitely many summands up to diffeomorphism, then M admits a complete metric of bounded geometry and uniformly positive scalar curvature. We show that the converse holds, generalising Perelman's theorem: Theorem 1.1. Let M be a connected, orientable 3-manifold which carries a complete riemannian metric of bounded geometry and uniformly positive scalar curvature. Then there is a finite collection F of spherical manifolds such that M is a connected sum of copies of S 2 × S 1 or members of F .

In fact, the collection F depends only on bounds on the geometry and a lower bound for the scalar curvature (cf. Corollary 11.1.)

Our main tool is R. Hamilton's Ricci flow. Let us give a brief review of the analytic theory of Ricci flow on complete manifolds. The basic short time existence result is due to W.-X. Shi [START_REF] Shi | Ricci deformation of the metric on complete noncompact Riemannian manifolds[END_REF]: if M is a 3-manifold and g 0 is a complete riemannian metric on M which has bounded sectional curvature, then there exists ε > 0 and a Ricci flow g(•) defined on [0, ε) such that g(0) = g 0 , and for each t, g(t) is also complete of bounded sectional curvature.

For brevity, we say that a Ricci flow g(•) has a given property P if for each time t, the riemannian metric g(t) has property P. Hence the solutions constructed by Shi are complete Ricci flows with bounded sectional curvature. This seems to be a natural setting for the analytical theory of Ricci flow. 2 Uniqueness of complete Ricci flows with bounded sectional curvature is due to B.-L. Chen and X.-P. Zhu [START_REF] Chen | Uniqueness of the Ricci flow on complete noncompact manifolds[END_REF].

We shall provide a variant of Perelman's surgery construction for Ricci flow, which has the advantage of being suitable for generalisations to open manifolds. Perelman's construction can be summarised as follows. Let M be a closed, orientable 3-manifold. Start with an arbitrary metric g 0 on M. Consider a maximal Ricci flow solution {g(t)} t∈[0,Tmax) with initial condition g 0 . If T max = +∞, there is nothing to do. Otherwise, one analyses the behaviour of g(t) as t goes to T max and finds an open subset Ω ⊂ M where a limiting metric can be obtained. If Ω is empty, then the construction stops. Otherwise the ends of Ω have a special geometry: they are so-called εhorns. Removing neighbourhoods of those ends and capping-off 3-balls with nearly standard geometry, one obtains a new closed, possibly disconnected riemannian 3-manifold. Then one restarts Ricci flow from this new metric and iterates the construction. In order to prove that finitely many surgeries occur in any compact time interval, Perelman makes crucial use of the finiteness of the volume of the various riemannian manifolds involved.

When trying to generalise this construction to open manifolds, one encounters several difficulties. First, the above-mentioned volume argument breaks down. Second, a singularity with Ω = M could occur, i.e. there may exist a complete Ricci flow with bounded sectional curvature defined on some interval [0, T ) and maximal among complete Ricci flows with bounded sectional curvature, such that when t tends to T , g(t) converges to, say, a metric of unbounded curvature ḡ. Then it is not known whether Ricci flow with initial condition ḡ exists at all, and even if it does, the usual tools like the maximum principle may no longer be available. One can imagine, for example, an infinite sequence of spheres of the same radius glued together by necks whose curvature is going to infinity. In this situation (M, ḡ) would have no horns to do surgery on.

In order to avoid those difficulties, we shall perform surgery before a singularity appears. To this end, we introduce a new parameter Θ, which determines when surgery must be done (namely when the supremum R max of the scalar curvature reaches Θ.) We do surgery on tubes rather than horns. Furthermore, we replace the volume argument for non-accumulation of surgeries by a curvature argument: the key point is that at each surgery time, R max drops by a definite factor (which for convenience we choose equal to 1/2.) This, together with an estimate on the rate of curvature blow-up, is sufficient to bound from below the elapsed time between two consecutive surgeries.

The idea of doing surgery before singularity time is not new: it was introduced by R. Hamilton in his paper [START_REF] Hamilton | Four-manifolds with positive isotropic curvature[END_REF] on 4-manifolds of positive isotropic curvature. Our construction should also be compared with that of G. Huisken and C. Sinestrari [START_REF] Huisken | Mean curvature flow with surgeries of two-convex hypersurfaces[END_REF] for Mean Curvature Flow, where in particular there is a similar argument for non-accumulation of surgeries. Needless to say, we rely heavily on Perelman's work, in particular the notions of κ-noncollapsing and canonical neighbourhoods.

Our construction should have other applications. In fact, it has already been adapted by H. Huang [START_REF] Huang | Complete 4-manifolds with uniformly positive isotropic curvature[END_REF] to complete 4-dimensional manifolds of positive isotropic curvature, using work of B.-L. Chen, S.-H. Tang and X.-P. Zhu [START_REF] Chen | Complete classification of compact four-manifolds with positive isotropic curvature[END_REF] in the compact case.

Remaining informal for the moment, we provisionally define a surgical solution as a sequence of Ricci flow solutions {(M i , g i (t))} t∈[t i ,t i+1 ] , with

0 = t 0 < • • • < t i < • • • +∞ is discrete in R, such that M i+1 is obtained from M i
by splitting along a locally finite collection of pairwise disjoint embedded 2-spheres, capping-off 3-balls and removing components which are spherical or diffeomorphic to R 3 , S 2 × S 1 , S 2 × R, RP 3 #RP 3 or a punctured RP 3 . If M i+1 is nonempty, we further require that R min (g i+1 ) R min (g i ) at time t i+1 . The formal definition of surgical solutions will be given in Section 2.

The components that are removed at time t i+1 are said to disappear. If all components disappear, that is if M i+1 = ∅, we shall say that the surgical solution becomes extinct at time t i+1 . In that case, it is straightforward to reconstruct the topology of the original manifold M 0 as a connected sum of the disappearing components (cf. Proposition 2.3 below.) Since R 3 , S 2 × R and punctured RP 3 's are themselves connected sums of spherical manifolds (in fact infinite copies of S 3 and RP 3 ,) the upshot is that M 0 is a connected sum of spherical manifolds and copies of S 2 × S 1 .

A simplified version of our main technical result follows.

Theorem 1.2. Let M be an orientable 3-manifold. Let g 0 be a complete riemannian metric on M which has bounded geometry. Then there exists a complete surgical solution of bounded geometry defined on [0, +∞), with initial condition (M, g 0 ).

When in addition we assume that g 0 has uniformly positive scalar curvature, we get (from the maximum principle and the condition that surgeries do not decrease R min ) an a priori lower bound for R min which goes to infinity in finite time. This implies that surgical solutions given by Theorem 1.2 are automatically extinct. As a consequence, any 3-manifold satisfying the hypotheses of Theorem 1.1 is a connected sum of spherical manifolds and copies of S 2 × S 1 . However, we also need to prove finiteness of the summands up to diffeomorphism. Below we state a more precise result, which will suffice for our needs.

We say that a riemannian metric g 1 is ε-homothetic to some riemannian metric g 2 if there exists λ > 0 such that λg 1 is ε-close to g 2 in the C [ε -1 ]+1topology. A riemannian metric which is ε-homothetic to a round metric (i.e. a metric of constant sectional curvature 1) is said to be ε-round.

Theorem 1.3. For all ρ 0 , T > 0 there exist Q, ρ > 0 such that if (M 0 , g 0 ) is a complete riemannian orientable 3-manifold which has sectional curvature bounded in absolute value by 1 and injectivity radius greater than or equal to ρ 0 , then there exists a complete surgical solution defined on [0, T ], with initial condition (M 0 , g 0 ), sectional curvature bounded in absolute value by Q and injectivity radius greater than or equal to ρ and such that all spherical disappearing components have scalar curvature at least 1, and are 10 -3 -round or diffeomorphic to S 3 or RP 3 .

Let us explain why this stronger conclusion implies that there are only finitely many disappearing components up to diffeomorphism. By definition, nonspherical disappearing components belong to a finite number of diffeomorphism classes. Now by the Bonnet-Myers theorem, 10 -3 -round components with scalar curvature at least 1 have diameter bounded above by some universal constant. Putting this together with the bounds on sectional curvatures and injectivity radius, the assertion then follows from Cheeger's finiteness theorem.

Remark that there is an apparent discrepancy between Theorems 1.2 and 1.3 in that in the former, the surgical solution is defined on [0, +∞) whereas in the latter it is only defined on a compact interval. However, Theorem 1.2 can be formally deduced from Theorem 1.3 via iteration and rescalings (cf. remark at the end of Section 2.)

Throughout the paper, we use the following convention:

All 3-manifolds considered here are orientable.

Here is a concise description of the content of the paper: in Section 2, we give some definitions, in particular the formal definition of surgical solutions, and show how to deduce Theorem 1.1 from Theorem 1.3. The remainder of the article (except the last section) is devoted to the proof of Theorem 1.3. In Section 3, we discuss Hamilton-Ivey curvature pinching, the standard solution, and prove the Metric Surgery Theorem, which allows to perform surgery. In Section 4, we recall some definitions and results on κ-noncollapsing, κ-solutions, and canonical neighbourhoods, and fix some constants that will appear throughout the rest of the proof.

In Section 5, we introduce the important notion of (r, δ, κ)-surgical solutions. These are special surgical solutions satisfying various estimates, and with surgery performed in a special way, according to the construction of Section 3. We state an existence theorem for those solutions, Theorem 5.3, which implies Theorem 1.3. Then we reduce Theorem 5.3 to three propositions, called A, B, and C. Sections 6 through 10 are devoted to the proofs of Propositions A, B, C, together with some technical results that are needed in these proofs.

Section 11 deals with generalisations of Theorem 1.3. One of them is an equivariant version, Theorem 11.2, which implies a classification of 3manifolds admitting metrics of uniformly positive scalar curvature whose universal cover has bounded geometry. We note that equivariant Ricci flow with surgery in the case of finite group actions on closed 3-manifolds has been studied by J. Dinkelbach and B. Leeb [START_REF] Dinkelbach | Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds[END_REF]. We follow in part their discussion; however, things are much simpler in our case, since we are mainly interested in the case of free actions. We also give a version of Theorem 5.3 with extra information on the long time behaviour. This may be useful for later applications. Finally, we review some global and local compactness results for Ricci flows in two appendices.

Acknowledgements

The authors wish to thank the Agence Nationale de la Recherche for its support under the programs F.O.G. (ANR-07-BLAN-0251-01) and GROUPES (ANR-07-BLAN-0141). The third author thanks the Institut de Recherche Mathématique Avancée, Strasbourg who appointed him while this work was done.

We warmly thank Joan Porti for numerous fruitful exchanges. The idea of generalising Perelman's work to open manifolds was prompted by conversations with O. Biquard and T. Delzant. We also thank Ch. Boehm, J. Dinkelbach, B. Leeb, T. Schick, B. Wilking and H. Weiss.

Surgical solutions

Let M be a possibly noncompact, possibly disconnected (orientable) 3-manifold.

Definitions

Definition. Let I ⊂ R be an interval. An evolving Riemannian manifold is a pair {(M(t), g(t))} t∈I where for each t, M(t) is a (possibly empty, possibly disconnected) manifold and g(t) a riemannian metric on M(t). We say that it is piecewise C 1 -smooth if there exists J ⊂ I, which is discrete as a subset of R, such that the following conditions are satisfied: i. On each connected component of I \ J, t → M(t) is constant, and

t → g(t) is C 1 -smooth.
ii. For each t 0 ∈ J, M(t 0 ) = M(t) for any t < t 0 sufficiently close to t 0 and t → g(t) is left continuous at t 0 .

iii. For each t 0 ∈ J \ {sup I}, t → (M(t), g(t)) has a right limit at t 0 , denoted (M

+ (t 0 ), g + (t 0 )) A time t ∈ I is regular if t has a neighbourhood in I where M(•) is constant and g(•) is C 1 -smooth. Otherwise it is singular. Definition. A piecewise C 1 -smooth evolving Riemannian 3-manifold {(M(t), g(t))} t∈I is a surgical solution of the Ricci Flow equation dg dt = -2 Ric g(t) (1)
if the following statements hold:

i. Equation ( 1) is satisfied at all regular times; ii. For each singular time t we have R min (g + (t)) ≥ R min (g(t));

iii. For each singular time t there is a locally finite collection S of disjoint embedded 2-spheres in M(t) and a manifold M ′ such that (a) M ′ is obtained from M(t) \ S by capping-off 3-balls;

(b) M + (t) is a union of connected components of M ′ and g(t) = g + (t) on M(t) ∩ M + (t); (c) Each component of M ′ \ M + (t) is spherical, or diffeomorphic to R 3 , S 2 × S 1 , S 2 × R, RP 3 #RP 3 or a punctured RP 3 . A component of M ′ \ M + (t) is said to disappear at time t. M (t) S 2 S 2 B 3 M (t 0 ) M ′ M + (t 0 ) disappearing components dg dt = -2 Ric g(t) dg dt = -2 Ric g(t)
An evolving riemannian manifold {(M(t), g(t))} t∈I is complete (resp. has bounded geometry) if for each t ∈ I such that M(t) = ∅, the riemannian manifold (M(t), g(t)) is complete (resp. has bounded geometry).

Deduction of Theorem 1.1 from Theorem 1.3

The purpose of this subsection is to explain how to deduce Theorem 1.1 from Theorem 1.3. For this, we need a result about the evolution of R min in a surgical solution. For convenience, we take the convention that R min (t) is +∞ if M(t) is empty.

Proposition 2.1. Let (M(•), g(•)) be a complete 3-dimensional surgical solution with bounded sectional curvature defined on an interval [0, T ). Assume that R min (0) ≥ 0. Then

R min (t) ≥ R min (0) 1 -2tR min (0)/3 .
Proof. Follows from the evolution equation for scalar curvature, the maximum principle for complete Ricci flows of bounded curvature [CLN06, Corollary 7.45] and the assumption that the minimum of scalar curvature is nondecreasing at singular times of surgical solutions.

Corollary 2.2. For every R 0 > 0 there exists T = T (R 0 ) such that the following holds. Let (M(•), g(•)) be a complete 3-dimensional surgical solution defined on [0, T ], with bounded sectional curvature, and such that R min (0) ≥ R 0 . Then (M(•), g(•)) is extinct.

We now recall the definition of capping-off 3-balls to a 3-manifold.

Definition. Let M, M ′ be 3-manifolds. Let S be a locally finite collection of embedded 2-spheres in M. One says that M ′ is obtained from M\S by capping-off 3-balls if there exists a collection B of 3-balls such that M ′ is the disjoint union

M ′ = M \ S B,
where each S ∈ S has a tubular neighbourhood V ⊂ M such that V \ S has two connected components V -, V + and there exists B -, B + ∈ B such that V -B -and

V + B + are 3-balls in M ′ . Conversely, each B ∈ B is included in such a 3-ball of M ′ .
Note that it is implicit in the above definition that there is an orientation preserving diffeomorphism, say φ -: ∂B -→ S ⊂ ∂V -, such that identifying ∂B -to the corresponding boundary of V -one obtains a 3-ball. From [START_REF] Smale | Diffeomorphisms of the 2-sphere[END_REF], the differentiable structure of M ′ does not depend of the above diffeomorphisms. Moreover, if M ′ and M ′′ are obtained from M \ S by capping off 3-balls, one can choose the diffeomorphism from M ′ to M ′′ to be the identity on

M ∩ M ′ = M ∩ M ′′ .
We shall need the following topological lemma:

Proposition 2.3. Let X be a class of closed 3-manifolds. Let M be a 3manifold. Suppose that there exists a finite sequence of 3-manifolds M 0 , M 1 , . . . , M p such that M 0 = M, M p = ∅, and for each i, M i is obtained from M i-1 by splitting along a locally finite collection of pairwise disjoint, embedded 2-spheres, capping off 3-balls, and removing some components which are connected sums of members of X . Then each component of M is a connected sum of members of X .

Proof. We prove the result by induction on p. The case p = 1 is immediate from the definition of a connected sum. Supposing that the proposition is true for some p, we consider a sequence M 0 , M 1 , . . . , M p+1 such that M 0 = M, M p+1 = ∅, and for each i, M i is obtained from M i-1 by splitting along a locally finite collection of 2-spheres, capping off 3-balls, and removing some components which are connected sums of members of X . By the induction hypothesis, M 1 is a connected sum of members of X .

Let S be the collection of 2-spheres involved in the process of turning M 0 into M 1 . Let B be the collection of capped-off 3-balls. Let S ′ be the collection of 2-spheres involved in the connected sum decomposition of M 1 . If B ∩ S ′ is empty, then the spheres of S ′ actually live in M 0 , and the union of S and S ′ splits M 0 into prime summands homeomorphic to members of X. This observation reduces our proof to the following claim:

Claim. S ′ can be made disjoint from B by an ambient isotopy.

Let us prove the claim. For each component B i of B, we fix a 3-ball B ′ i contained in the interior of B i and disjoint from S ′ , and a collar neighbourhood U i of ∂B i in M 1 \ B i . Since {B i } is locally finite, we may ensure that the U i 's are pairwise disjoint. Choose an ambient isotopy of M 1 which takes B ′ i onto B i and B i onto B i ∪ U i for each i. Then after this ambient isotopy, S ′ is still locally finite, and is now disjoint from B.

To see why these results imply Theorem 1.1, take a 3-manifold M and a complete metric g 0 of bounded geometry and uniformly positive scalar curvature on M. By rescaling if necessary we can assume that the bound on the curvature is 1. From the positive lower bound on R min (g 0 ) we get an a priori upper bound T for the extinction time of a surgical solution, using Corollary 2.2. Applying Theorem 1.3, we get numbers Q, ρ and a surgical solution (M(•), g(•)) with initial condition (M, g 0 ) defined on [0, T ] satisfying the two additional conditions. This solution is extinct, and as we have already explained in the introduction, the disappearing components are connected sums of spherical manifolds and copies of S 2 × S 1 , the summands belonging to some finite collection which depends only on the bounds on the geometry. Let X be the collection of prime factors of the disappearing components. Let 0 = t 0 < t 1 < t 2 < • • • < t p = T be a set of regular times of (M(•), g(•)) such that there is exactly one singular time between each pair of consecutive t i 's. The conclusion of Theorem 1.1 now follows from Proposition 2.3 applied with M i = M(t i ).

More definitions

Notation Let n ≥ 2 be an integer and (M, g) be a riemannian n-manifold.

For any x ∈ M, we denote by Rm(x) : Λ 2 T x M → Λ 2 T x M the curvature operator, and | Rm(x)| its norm, which is also the maximum of the absolute values of the sectional curvatures at x. We let R(x) denote the scalar curvature of x. The infimum (resp. supremum) of the scalar curvature of g on M is denoted by R min (g) (resp. R max (g)).

We write d : M × M → [0, ∞) for the distance function associated to g. For r > 0 we denote by B(x, r) the open ball of radius r around x. Finally, if x, y are points of M, we denote by [xy] a geodesic segment connecting x to y. This is a (common) abuse of notation, since such a segment is not unique in general.

For closeness of metrics we adopt the conventions of [BBB + , Section 0.6.1].

Let {(M(t), g(t))} t∈I be a surgical solution and t ∈ I. If t is singular, one sets M reg (t) := M(t) ∩ M + (t) and denotes by M sing (t) its complement, i.e. M sing (t) := M(t) \ M reg (t) = M(t) \ M + (t). If t is regular, then M reg (t) = M(t) and M sing (t) = ∅.

At a singular time, connected components of M sing (t) belong to three types: i. components of M(t) which are disappearing components of M ′ , ii. closures of components of M(t) \ S which give, after being capped-off, disappearing components of M ′ , and iii. embedded 2-spheres of S.

In particular, the boundary of M sing (t) is contained in S.

We say that a pair (x, t) ∈ M × I is singular if x ∈ M sing (t); otherwise we call (x, t) regular.

Definition. Let t 0 be a time, [a, b] be an interval containing t 0 and X be a subset of M(t 0 ) such that for every t ∈ [a, b), we have X ⊂ M reg (t). Then the set X × [a, b] is unscathed. Otherwise, we say that X is scathed.

Remarks.

1) In the definition of 'unscathed', we allow the final time slice to contain singular points, i.e. we may have Notation For t ∈ I and x ∈ M(t) we use the notation Rm(x, t), R(x, t) to denote the curvature operator and the scalar curvature respectively. For brevity we set R min (t) := R min (g(t)) and R max (t) := R max (g(t)).

X ∩ M sing (b) = ∅. The point is that if X × [a,
We use d t (•, •) for the distance function associated to g(t). The ball of radius ρ around x for g(t) is denoted by B(x, t, ρ).

For the definition of closeness of evolving riemannian manifolds, we refer to [BBB + , Section 0.6.2.].

Definition. Let t 0 ∈ I and Q > 0. The parabolic rescaling with factor Q at time t 0 is the evolving manifold {( M (t), ḡ(t))} where M (t) = M(t 0 + t/Q), and ḡ(t

) = Q g(t 0 + t Q ).
Finally we remark that Theorem 1.2 follows by iteration of Theorem 1.3 via parabolic rescalings. Hence in the sequel, we focus on proving Theorem 1.3.

Metric surgery

Curvature pinched toward positive

Let (M, g) be a 3-manifold and x ∈ M be a point. We denote by λ(x) ≥ µ(x) ≥ ν(x) the eigenvalues of the curvature operator Rm(x). By our definition, all sectional curvatures lie in the interval [ν(x), λ(x)]. Moreover, λ(x) (resp. ν(x)) is the maximal (resp. minimal) sectional curvature at x. If C is a real number, we sometimes write Rm(x)

≥ C instead of ν(x) ≥ C. Likewise, Rm(x) ≤ C means λ(x) ≤ C.
It follows that the eigenvalues of the Ricci tensor are equal to λ + µ, λ + ν, and µ + ν; as a consequence, the scalar curvature R(x) is equal to 2(λ(x) + µ(x) + ν(x)).

For evolving metrics, we use the notation λ(x, t), µ(x, t), and ν(x, t), and correspondingly write Rm(x, t) ≥ C for ν(x, t) ≥ C, and Rm(x, t) ≤ C for λ(x, t) ≤ C.

Let φ be a nonnegative function. A metric g on M has φ-almost nonnegative curvature if Rm ≥ -φ(R).

Now we consider a familly of positive functions (φ t ) t 0 defined as follows. Set s t := e 2 1+t and define φ t : [-2s t , +∞) -→ [s t , +∞) as the reciprocal of the function s → 2s(ln(s) + ln(1 + t) -3).

Following [START_REF] Morgan | Ricci Flow and the Poincaré Conjecture, volume 3 of Clay mathematics monographs[END_REF], we use the following definition.

Definition. Let I ⊂ [0, ∞) be an interval, t 0 ∈ I and {g(t)} t∈I be an evolving metric on M. We say that g(•) has curvature pinched toward positive at time

t 0 if for all x ∈ M we have R(x, t 0 ) - 6 4t 0 + 1 , (2)
Rm(x, t 0 ) -φ t 0 (R(x, t 0 )). (3)
We say that g(•) has curvature pinched toward positive if it has curvature pinched toward positive at each t ∈ I. The following easy lemmas will be useful.

Remark that if | Rm(g(0))| ≤ 1,
Lemma 3.2. i) φ t (s) = φ 0 ((1+t)s) 1+t . ii) φt(s) s decreases to 0 as s tends to +∞. iii) φ 0 (s) s = 1 4 if s = 4e 5 . Proof. [BBB + , Lemma 2.4.6]
The main purpose of Property ii) is to ensure that limits of suitably rescaled evolving metrics with curvature pinched toward positive have nonnegative curvature operator (see Proposition 6.4). In the sequel we set s := 4e 5 . Lemma 3.3 (Pinching Lemma). Assume that g(•) has curvature pinched toward positive and let t 0, r > 0 be such that

(1 + t)r -2 s. If R(x, t) r -2 then | Rm(x, t)| r -2 . Proof. [BBB + , Lemma 2.4.7]

The standard solution

We recall the definition we used in [BBB + ] as initial condition of the standard solution. The functions f, u below are chosen in [BBB + , Section 5.1].

Definition. Let dθ 2 denote the round metric of scalar curvature 1 on S 2 .

The initial condition of the standard solution is the riemannian manifold S 0 = (R 3 , ḡ0 ), where the metric ḡ0 is given in polar coordinates by: ḡ0 = e -2f (r) g u , where g u = dr 2 + u(r) 2 dθ 2 .

We also define S u := (R 3 , g u ). The origin of R 3 , which is also the centre of spherical symmetry, will be denoted by p 0 .

In particular, (B(0, 5), ḡ0 ) has positive sectional curvatures (see [BBB + , Lemma 5.1.2]), and the complement of B(0, 5) is isometric to S 2 × [0, +∞).

Ricci flow with initial condition S 0 has a maximal solution defined on [0, 1), which is unique among complete flows of bounded sectional curvature [START_REF] Perelman | Ricci flow with surgery on three-manifolds[END_REF]. This solution is called the standard solution.

The standard ε-neck is the riemannian product S 2 × (-ε -1 , ε -1 ), where the S 2 factor is round of scalar curvature 1. Its metric is denoted by g cyl . We fix a basepoint * in S 2 × {0}.

Definition. Let (M, g) be a riemannian 3-manifold and x be a point of M.

A neighbourhood N ⊂ M of x is called an ε-neck centred at x if (N, g, x) is ε-homothetic to (S 2 × (-ε -1 , ε -1 ), g cyl , * ).
If N is an ε-neck and ψ : N ε → N is a parametrisation, i.e. a diffeomorphism such that some rescaling of ψ * (g) is ε-close to g cyl , then the sphere ψ(S 2 × {0}) is called a middle sphere of N.

Definition. Let δ, δ ′ be positive numbers. Let g be a riemannian metric on M. Let (U, V, p, y) be a 4-tuple such that U is an open subset of M, V is a compact subset of U, p ∈ IntV , y ∈ ∂V . Then (U, V, p, y) is called a marked (δ, δ ′ )-almost standard cap if there exists a δ ′ -isometry ψ : B(p 0 , 5 + δ -1 ) → (U, R(y)g), sending B(p 0 , 5) to IntV and p 0 to p. One calls V the core and p the tip.

The metric surgery theorem

Theorem 3.4 (Metric surgery). There exist δ 0 > 0 and a function δ ′ : (0, δ 0 ] ∋ δ → δ ′ (δ) ∈ (0, ε 0 /10] tending to zero as δ → 0, with the following property:

Let φ be a nondecreasing, positive function; let δ ≤ δ 0 ; let (M, g) be a riemannian 3-manifold with φ-almost nonnegative curvature, and {N i } be a locally finite collection of pairwise disjoint δ-necks in M. Let M ′ be a manifold obtained by cutting M along the middle spheres of the N i 's and capping off 3-balls.

Then there exists a riemannian metric g + on M ′ such that:

i. g + = g on M ′ ∩ M;
ii. For each component B of M ′ \ M, there exist p ∈ IntB and y ∈ ∂B such that (N ′ ∪ B, B, p, y) is a marked (δ, δ ′ (δ))-almost standard cap with respect to g + , where N ′ is the 'half ' of N adjacent to B in M ′ ;

iii. g + has φ-almost nonnegative curvature.

Remark. In the application of the above theorem, M + will be a submanifold of M ′ .

Proof. On M ′ ∩ M we set g + := g. On the added 3-balls we define g + as follows. Let N ⊂ M be one of the δ-necks of the collection, and let S be its middle sphere. By definition there exists a diffeomorphism ψ :

S 2 × (-δ -1 , δ -1 ) -→ N and a real number λ > 0 such that ||ψ * λg -g cyl || < δ in the C [δ -1 ]+1 -norm (see [BBB + ]
for the details). Note that for each y ∈ N we have that λ/R(y) is δ ′ -close to 1 for some universal δ ′ (δ) tending to zero with δ.

Define N + := ψ(S 2 × [0, δ -1 )), i.e. N + is the right half of the neck. Let Σ ⊂ M ′ \ M be the 3-ball that is capped-off to it and Φ : ∂Σ -→ ∂N + be the corresponding diffeomorphism. Our goal is to define g + on Σ in such a way that (N

+ ∪ Φ Σ, g + ) is a (δ, δ ′ (δ))-almost standard cap with φ-almost nonnegative curvature.
Let us introduce more notation. For 0 ≤ r 1 ≤ r 2 , we let C[r 1 , r 2 ] denote the annular region of R 3 defined by the inequations r 1 ≤ r ≤ r 2 in polar coordinates. Observe that for all 3 ≤ r 1 < r 2 , the restriction of g u to C[r 1 , r 2 ] is isometric to the cylinder S 2 × [r 1 , r 2 ] with scalar curvature 1. We consider

B := B(0, 5) ⊂ S u . Set V -:= ψ(S 2 × (-2, 0]) and V + := ψ(S 2 × [0, 2)). Restrict ψ on S 2 × (-2, δ -1 ) to V -∪ N + , where S 2 × (-2, δ -1 ) is now considered as the annulus C(3, 5 + δ -1 ) ⊂ S u . N S u 0 N + N - 3 5 7 5 + δ -1 ψ V -V +

S

Let ḡ := ψ * (λg) be the pulled-back rescaled metric on C(3, 5 + δ -1 ). Note that ||ḡg u || < δ on this set and that ḡ has φ-almost nonnegative curvature.

On B(0, 5 + δ -1 ) we define in polar coordinates

ḡ+ := e -2f (χg u + (1 -χ)ḡ) = χḡ 0 + (1 -χ)e -2f ḡ where χ : [0, 5 + δ -1 ] → [0, 1] is a smooth function satisfying    χ ≡ 1 on [0, 3] χ ′ < 0 on (3, 4) χ ≡ 0 on [4, 5 + δ -1 ]. Note that    ḡ+ = ḡ0 on B(0, 3) ḡ+ = e -2f ḡ on C[4, 5] ḡ+ = ḡ on C[5, 5 + δ -1 ).
Finally set

g + := (ψ -1 ) * (λ -1 ḡ) = g on N + g + := λ -1 ḡ+ on B(0, 5). N 0 S N + 3 5 7 ψ ḡ V + ḡ0 ḡ (B(0, 5 + δ -1 ), ḡ+ )
Let p be the origin and y be an arbitrary point of ∂B. There remains to show that ((N + ∪ ψ |∂B B, B, p, y), g + ) is a (δ, δ ′ (δ))-almost standard cap, and has φ-almost nonnegative curvature. It suffices clearly to consider g + on B, or ḡ+ on B(0, 5). This is tackled by the following proposition ([BBB + , Proposition 5.2.2]), applied to ḡ with the rescaled pinching function s → λ -1 φ(λs): Proposition 3.5. There exists δ 1 > 0 and a function δ ′ : (0, δ 1 ] -→ (0, ε 0 10 ] with limit zero at zero, having the following property: let φ be a nondecreasing positive function, 0 < δ ≤ δ 1 and ḡ be a metric on

C(3, 5) ⊂ R 3 , with φ- almost nonnegative curvature, such that ||ḡ -g u || C [ 1 δ ]+1 < δ on C(3, 5). Then the metric ḡ+ = e -2f (χg u + (1 -χ)ḡ)
has φ-almost nonnegative curvature, and is δ ′ (δ)-close to ḡ0 on B(0, 5).

Setting δ 0 := δ 1 completes the proof of Theorem 3.4.

κ-noncollapsing and canonical neighbourhoods 4.1 κ-noncollapsing

Let {(M(t), g(t))} t∈I be an evolving riemannian manifold. We say that a pair (x, t) is a point in spacetime if t ∈ I and x ∈ M(t). For convenience we denote by M the set of all such points. A (backward) parabolic neighbourhood of a point (x, t) in spacetime is a set of the form

P (x, t, r, -∆t) := {(x ′ , t ′ ) ∈ M | x ′ ∈ B(x, t, r), t ′ ∈ [t -∆t, t]}.
In particular, the set P (x, t, r, -r 2 ) is called a parabolic ball of radius r.

A parabolic neighbourhood P (x, t, r, -∆t) is unscathed if B(x, t, r) × [t - ∆t, t] is unscathed. In this case P (x, t, r, -∆t) = B(x, t, r) × [t -∆t, t].
Definition. Fix κ, r > 0. We say that (M(•), g(•)) is κ-collapsed at (x, t) on the scale r if for all (x ′ , t ′ ) ∈ P (x, t, r, -r 2 ) one has | Rm(x ′ , t ′ )| ≤ r -2 , and volB(x, t, r) < κr n . Otherwise, (M(•), g(•)) is κ-noncollapsed at (x, t) on the scale r.

We say that (M(•), g(•)) is κ-noncollapsed on the scale r if it is κ-noncollapsed on this scale at every point of M.

Canonical neighbourhoods

Definition. Let (M, g) be a riemannian 3-manifold and x be a point of M. We say that U is an ε-cap centred at

x if U is the union of two sets V, W such that x ∈ IntV , V is diffeomorphic to B 3 or RP 3 \ B 3 , W ∩ V = ∂V , and W is an ε-neck. A subset V as above is called a core of U. Let ε > 0, C >> ε -1 , {(M(t), g(t)
)} t∈I be an evolving riemannian manifold and (x 0 , t 0 ) be a point in spacetime.

We call cylindrical flow the manifold S 2 × R together with the product flow on (-∞, 0], where the first factor is round, normalised so that the scalar curvature at time 0 is identically 1. We denote this evolving metric by g cyl (t).

Definition. An open subset N ⊂ M(t 0 ) is called a strong ε-neck3 centred at (x 0 , t 0 ) if there is a number Q > 0 such that (N, {g(t)} t∈[t 0 -Q -1 ,t 0 ] , x 0 ) is unscathed, and after parabolic rescaling with factor Q at time t 0 , ε-close to (S 2 × (-ε -1 , ε -1 ), {g cyl (t)} t∈[-1,0] , * ).

Remark. Let Q > 0, and consider the flow (S 2 × R, Qg cyl (tQ -1 )) restricted to (-Q, 0]. Then for every x ∈ S 2 × R and every ε > 0, the point (x, 0) is centre of a strong ε-neck.

We recall that ε-round means ε-homothetic to a metric of positive constant sectional curvature.

Definition. Let U ⊂ M(t) be an open subset and x ∈ U such that R(x) := R(x, t) > 0. One says that U is an (ε, C)-canonical neighbourhood centred at (x, t) if: A. U is of a strong ε-neck centred at (x, t), or B. U is an ε-cap centred at x for g(t), or C. (U, g(t)) has sectional curvatures > C -1 R(x) and is diffeomorphic to S 3 or RP 3 , or D. (U, g(t)) is ε-round,
and if moreover, the following estimates hold in cases A, B, C for (U, g(t)):

There exists r ∈ (C -1 R(x) -1 2 , CR(x) -1 2 ) such that: i. B(x, r) ⊂ U ⊂ B(x, 2r);
ii. The scalar curvature function restricted to U has values in a compact subinterval of (C -1 R(x), CR(x));

iii. If B(y, r) ⊂ U and if | Rm | ≤ r -2 on B(y, r) then C -1 < volB(y, r) r 3 ; (4) iv. |∇R(x)| < CR(x) 3 2 , (5) v
. |∆R(x) + 2| Ric(x)| 2 | < CR(x) 2 , (6) vi
. |∇ Rm(x)| < C| Rm(x)| 3 2 , (7)
Remarks.

• In case D, Estimates (i)-(vi) hold except maybe (iii) (consider e.g. lens spaces.)

• (i) implies that diam U is at most 4r, which in turn is bounded above by a function of C and R(x).

• (iii) implies that vol U is bounded from below by C -1 R(x) -3/2 .
• Estimate (v) implies the following scale-invariant bound on the timederivative of R (at a regular time):

| ∂R ∂t (x, t)| < CR(x, t) 2 . ( 8
)
• We call (ε,C)-cap any ε-cap of (M, g) which satisfies (i)-(vi),

• In cases C and D, U is diffeomorphic to a spherical manifold.

• Cases C and D are not mutually exclusive.

• Being the centre of an (ε, C)-canonical neighbourhood is an open property in spacetime: if U ⊂ M(t) is unscathed on (tα, t + α) for some α > 0, then there exists a neighbourhood Ω of (x, t) such that any (x ′ , t ′ ) ∈ Ω is centre of an (ε, C)-canonical neighbourhood. In case A, one can use the same set N = U and factor Q, but change the parametrisation so that the basepoint * is sent to x rather than x 0 . Case B is similar. Cases C, D are obvious.

• The same argument shows that being the centre of an (ε, C)-canonical neighbourhood is also an open property with respect to a change of metric in the C [ε -1 ] -topology.

Fixing the constants

In order to fix the constants, we recall some results of Perelman on κ-solutions and the standard solution.

Theorem 4.1. For all ε > 0 there exists

C sol = C sol (ε) such that if (M, {g(t)} t∈(-∞,0] ) is a 3-dimensional κ-solution, then every (x, t) ∈ M × (-∞, 0] is centre of an (ε, C sol )-canonical neighbourhood.
Proposition 4.2. There exists κ st > 0 such that the standard solution is κ st -noncollapsed on all scales.

Proposition 4.3. For every ε > 0 there exists C st (ε) > 0 such that if (x, t) is a point in the standard solution such that t > 3/4 or x ∈ B(p 0 , 0, ε -1 ), then (x, t) has an (ε, C st )-canonical neighbourhood. Moreover there is an estimate R min (t) const st (1t) -1 for some constant const st > 0.

Next we recall two technical lemmas from [BBB + ] which allow to fix universal constants ε 0 and β 0 .

Lemma 4.4. There exists ε 0 > 0 such that the following holds. Let ε ∈ (0, 2ε 0 ]. Let (M, g) be a riemannian 3-manifold. Let y 1 , y 2 be points of M. Let U 1 ⊂ M be an ε-neck centred at y 1 with parametrisation ψ 1 : S 2 × (-ε -1 , ε -1 ) → U 1 and middle sphere S 1 . Let U 2 ⊂ M be a 10ε-neck centred at y 2 with middle sphere S 2 . Call π :

U 1 → (-ε -1 , ε -1 ) the composition of ψ -1
1 with the projection of S 2 × (-ε -1 , ε -1 ) onto its second factor. Assume that y 2 ∈ U 1 and |π(y 2 )| ≤ (2ε) -1 . Then the following conclusions hold:

i. U 2 is contained in U 1 ;
ii. The boundary components of ∂U 2 can be denoted by S + , S -in such a way that

π(S -) ⊂ [π(y 2 ) -(10ε) -1 -10, π(y 2 ) -(10ε) -1 + 10] ,
and

π(S + ) ⊂ [π(y 2 ) + (10ε) -1 -10, π(y 2 ) + (10ε) -1 + 10] ;
iii. The spheres S 1 , S 2 are isotopic in U 1 .

Proof.

[BBB + , Lemma 1.2.1]
Let K st be the supremum of the sectional curvatures of the standard solution on [0, 4/5].

Lemma 4.5. For all ε ∈ (0, 10 -4 ) there exists β = β(ε) ∈ (0, 1) such that the following holds.

Let a, b be real numbers satisfying a < b < 0 and |b| ≤ 3/4, let (M, g(•)) be a surgical solution defined on (a, 0], and x ∈ M be a point such that:

• R(x, b) = 1; • (x, b) is centre of a strong βε-neck; • P (x, b, (βε) -1 , |b|) is unscathed and satisfies | Rm | ≤ 2K st .
Then (x, 0) is centre of a strong ε-neck.

Proof. We can argue exactly as in [BBB + , Lemma 2.3.5.] Fix ε 0 > 0 so that Lemma 4.4 holds. Let β := β(ε 0 ) be the constant given by Lemma 4.5. Finally, define C 0 := max(100, 2C sol (ε 0 /2), 2C st (βε 0 /2)).

Definition. Let r > 0. An evolving riemannian manifold {(M(t), g(t))} t∈I has property (CN) r if for all (x, t) ∈ M, if R(x, t) ≥ r -2 , then (x, t) admits an (ε 0 , C 0 )-canonical neighbourhood.
Definition. Let κ > 0. An evolving riemannian manifold {(M(t), g(t))} t∈I has property (NC) κ if it is κ-noncollapsed on all scales less than 1. 5 (r, δ, κ)-surgical solutions 5.1 Cutoff parameters and (r, δ)-surgery Theorem 5.1 (Cutoff parameters). For all r, δ > 0, there exist h ∈ (0, δr) and D > 10 such that if (M(•), g(•)) is a complete surgical solution of bounded curvature defined on an interval [a, b], with curvature pinched toward positive and satisfying (CN) r , then the following holds:

Let t ∈ [a, b] and x, y, z ∈ M(t) such that R(x, t) ≤ 2/r 2 , R(y, t) = h -2
, and R(z, t) ≥ D/h 2 . Assume there is a curve γ connecting x to z via y, such that each point of γ with scalar curvature in

[2C 0 r -2 , C 0 -1 Dh -2 ] is centre of an ε 0 -neck. Then (y, t) is centre of a strong δ-neck.
This will be proved in Section 6. In the sequel we fix functions (r, δ) → h(r, δ) and (r, δ) → D(r, δ) with the above property. We set Θ := 2Dh -2 . This number will be used as a curvature threshold for the surgery process.

Definition. We say that two real numbers r, δ are surgery parameters if 0 < r < 10 -3 and 0 < δ < min(ε 0 , δ 0 ). The associated cutoff parameters are h := h(r, δ), D := D(r, δ) and Θ := 2Dh -2 .

From now on, we fix a function δ ′ : (0, δ 0 ] -→ (0, ε 0 /10] as in the metric surgery theorem. A marked (δ, δ ′ (δ))-almost standard cap will be simply called a δ-almost standard cap. An open subset U of M is called a δ-almost standard cap if there exist V , p and y such that (U, V, p, y) is a δ-almost standard cap.

Definition. Fix surgery parameters r, δ and let h, D, Θ be the associated cutoff parameters. Let {(M(t), g(t))} t∈I be an evolving riemannian manifold. Let t 0 ∈ I and (M + , g + ) be a (possibly empty) riemannian manifold. We say that (M + , g + ) is obtained from (M(•), g(•)) by (r, δ)-surgery at time t 0 if the following conditions are satisfied: i. M + is obtained from M(t 0 ) by cutting along a locally finite collection of disjoint 2-spheres, capping off 3-balls, and possibly removing some components that are spherical or diffeomorphic to R 3 , S 2 × R, RP 3 \ {pt}, RP 3 #RP 3 , S 2 × S 1 . In addition, a spherical manifold U can only be removed if it is contained in M(t 0 ), and (U, g(t 0 )) is ε-round and satisfies R ≥ 1;

ii. For all x ∈ M + \ M(t 0 ), there exists a δ-almost standard cap (U, V, p, y) in M + , such that

(a) x ∈ V ; (b) y ∈ M(t 0 ); (c) R(y, t 0 ) = h -2 ; (d) (y, t 0 ) is centre of a strong δ-neck.
iii. R max (g(t 0 )) = Θ, and if

M + = ∅, then R max (g + ) ≤ Θ/2.
Definition. Fix surgery parameters r, δ and let h, D, Θ be the associated cutoff parameters. Let I ⊂ [0, ∞) be an interval and {(M(t), g(t))} t∈I be a surgical solution. We say that it is an (r, δ)-surgical solution if it has the following properties:

i. It has curvature pinched toward positive and satisfies R(x, t) ≤ Θ for all (x, t) ∈ M;

ii. For every singular time

t 0 ∈ I, (M + (t 0 ), g + (t 0 )) is obtained from (M(•), g(•)
) by (r, δ)-surgery at time t 0 ;

iii. Condition (CN) r holds.

Let κ > 0. An (r, δ)-surgical solution which in addition satisfies Condition (NC) κ will be called an (r, δ, κ)-surgical solution.

Existence theorem for (r, δ, κ)-surgical solutions

Theorem 1.3 is implied by the following result: Theorem 5.2. For every ρ 0 > 0 and T ≥ 0, there exist r, δ, κ > 0 such that for any complete riemannian 3-manifold (M 0 , g 0 ) with | Rm | ≤ 1 and injectivity radius at least ρ 0 , there exists an (r, δ, κ)-surgical solution defined on [0, T ] satisfying the initial condition (M(0), g(0)) = (M 0 , g 0 ).

Theorem 5.2 is itself a special case of the following result, which has the advantage of being suitable for iteration: Theorem 5.3. For every Q 0 , ρ 0 > 0 and all 0 ≤ T A < T Ω , there exist r, δ, κ > 0 such that for any complete riemannian 3-manifold (M 0 , g 0 ) which satisfies | Rm | ≤ Q 0 , has injectivity radius at least ρ 0 , has curvature pinched toward positive at time T A , there exists an (r, δ, κ)-surgical solution defined on [T A , T Ω ] satisfying the initial condition (M(T A ), g(T A )) = (M 0 , g 0 ).

Note that in the statement of Theorem 5.2 the assumption of almost nonnegative curvature is not necessary since it is automatic. We shall prove Theorem 5.3 directly.

Our next aim is to reduce Theorem 5.3 to three results, called Propositions A, B, C, which are independent of one another.

Proposition A. There exists a universal constant δA > 0 having the following property: let r, δ be surgery parameters, a, b be positive numbers with a < b, and {(M(t), g(t))} t∈(a,b] be an (r, δ)-surgical solution. Suppose that δ ≤ δA , and R max (b) = Θ.

Then there exists a riemannian manifold (M + , g + ), which is obtained from (M(•), g(•)) by (r, δ)-surgery at time b, and in addition satisfies:

i. g + has φ b -almost nonnegative curvature; ii. R min (g + ) ≥ R min (g(b)).
Remark. The manifold M + may be empty. In this case, the second assertion in the conclusion follows from the convention R min (∅) = +∞.

Proposition B. For all Q 0 , ρ 0 , κ > 0 there exist r = r(Q 0 , ρ 0 , κ) < 10 -3 and δB = δB (Q 0 , ρ 0 , κ) > 0 with the following property: let δ ≤ δB , 0 ≤ T A < b and (M(•), g(•)) be a surgical solution defined on [T A , b] such that g(T A ) satisfies | Rm | ≤ Q 0 and has injectivity radius at least ρ 0 .

Assume that (M(•), g(•)) satisfies Condition (NC) κ/16 , has curvature pinched toward positive, and that for each singular time t 0 , (M + (t 0 ), g + (t 0 )) is obtained from (M(•), g(•)) by (r, δ)-surgery at time t 0 .

Then (M(•), g(•)) satisfies Condition (CN) r .

Proposition C. For all Q 0 , ρ 0 > 0 and all 0 ≤ T A < T Ω there exists κ = κ(Q 0 , ρ 0 , T A , T Ω ) such that for all 0 < r < 10 -3 there exists δC = δC (Q 0 , ρ 0 , T A , T Ω , r) > 0 such that the following holds. For all 0 < δ ≤ δC and b ∈ (T A , T Ω ], every (r, δ)-surgical solution defined on [T A , b) such that g(T A ) satisfies | Rm | ≤ Q 0 and has injectivity radius at least ρ 0 , satisfies (NC) κ .

Remark.

The formulation of Proposition B, and its use below, are somewhat different in [BBB + ]. In the compact case, it is fairly easy to prove that the (CN) r property is open with respect to time (see [BBB + , Lemma 3.3.2]). This is not the case here.

Proof of Theorem 5.3 assuming Propositions A, B, C. We start with two easy lemmas. The first one allows to control the density of surgery times by the surgery parameters.

Lemma 5.4. Let r, δ be surgery parameters. Let {(M(t), g(t))} t∈I be an (r, δ)-surgical solution. Let t 1 < t 2 be two singular times. Then t 2t 1 is bounded from below by a positive number depending only on r, δ.

Proof. We can suppose that M(•) is constant and g(•) is smooth on (t 1 , t 2 ]. Since R max (t 2 ) = Θ we can choose a point x ∈ M(t 2 ) such that R(x, t 2 ) ≥ R max (t 2 ) -1. Since R max (g + (t 1 )) ≤ Θ/2, there exists t + ∈ [t 1 , t 2 ] maximal such that lim t→t + ,t>t + R(x, t) = Θ/2. In particular, (x, t) admits an (ε 0 , C 0 )canonical neighbourhood for all t ∈ (t + , t 2 ]. Integrating inequality (8) on (t + , t 2 ] gives a positive lower bound for t 2t + depending only on Θ, hence only on r, δ.

The second one says that (NC) κ is a closed condition: Lemma 5.5. Let (M(•), g(•)) be a surgical solution defined on an interval (a, b], x ∈ M(b) and r, κ > 0. Suppose that for all t ∈ (a, b), x ∈ M(t), and (M(•), g(•)) is κ-noncollapsed at (x, t) on all scales less than or equal to r. Then it is κ-noncollapsed at (x, b) on the scale r.

Proof. [BBB + , Lemma 2.1.5.] Let Q 0 , ρ 0 > 0 and 0 ≤ T A < T Ω . Proposition C gives a constant κ = κ(Q 0 , ρ 0 , T A , T Ω ). Proposition B gives constants r, δB depending on κ. We can assume r -2 > 12Q 0 . Then apply Proposition C again to get a constant δC . Set δ = min( δA , δB , δC ). Without loss of generality, we assume that κ ≤ κ st .

From r, δ we get the cutoff parameters h, D, Θ. Let (M 0 , g 0 ) be a riemannian manifold which has φ A -almost nonnegative curvature, satisfies R min (g 0 ) ≥ -6/(4T A +1), | Rm | ≤ Q 0 , and has injectivity radius at least ρ 0 .

Let X be the set of ordered pairs (b, {(M(t), g(t))} t∈[T A ,b) ) consisting of a number b ∈ (T A , T Ω ] and an (r, δ, κ)-surgical solution such that (M(T A ), g(T A )) = (M 0 , g 0 ). We first show that X is nonempty. By standard results on the Ricci flow (see e.g. [CLN06, Lemma 6.1]) there exists a complete Ricci flow solution g(•) defined on [T A , T A +(16Q 0 ) -1 ], such that g(T A ) = g 0 and | Rm | 2Q 0 on the interval. By Proposition 3.1, g(•) has curvature pinched toward positive on

[T A , T A + (16Q 0 ) -1 ]. As R 12Q 0 < r -2 < Θ, g(•)
satisfies Property (i) of an (r, δ)-surgical solution, and Properties (ii), (iii) are vacuously satisfied. By Proposition C, it satisfies (NC) κ on the interval. Hence

g(•) is a (r, δ, κ)-surgical solution on [T A , T A + (16Q 0 ) -1 ]. The set X has a partial ordering, defined by (b 1 , (M 1 (•), g 1 (•))) ≤ (b 2 , (M 2 (•), g 2 (•))) if b 1 ≤ b 2 and (M 2 (•), g 2 (•)) is an extension of (M 1 (•), g 1 (•)).
We want to use Zorn's lemma to prove existence of a maximal element in X . In order to do this, we consider an infinite chain, i. e. an infinite sequence of numbers

T A < b 1 < b 2 < • • • b n < • • • < T Ω
and of (r, δ, κ)-surgical solutions defined on the intervals [T A , b n ), and which extend one another. In this way we get an evolving manifold {(M(t), g(t))} defined on [T A , b ∞ ), where b ∞ is the supremum of the b n 's. By Lemma 5.4, the set of singular times is a discrete subset of R, so {(M(t), g(t))} t∈[T A ,b∞) is an (r, δ, κ)-surgical solution, thus a majorant of the increasing sequence.

Hence we can apply Zorn's lemma. Let (b max , (M(•), g(•))) ∈ X be a maximal element. Its scalar curvature lies between -6 and Θ, so it is bounded independently of t. Its curvature is pinched toward positive so the sectional curvature is also bounded independently of t. Using the Shi estimates, we deduce that all derivatives of the curvature are also bounded at time b max . This allows to take a smooth limit and extend (M(•), To conclude, we prove by contradiction that b max = T Ω . Assume that b max < T Ω and consider the following two cases:

Case 1 R max (b max ) < Θ.
Applying the short time existence theorem for Ricci flow with initial metric g(b max ), we can extend g(•) to a surgical solution defined on an interval [T A , b max + α) for some α > 0. We choose α sufficiently small so that we still have R max (t) < Θ on [T A , b max + α). There are no singular times in [b max , b max + α), and by Proposition 3.1 the extension satisfies the hypothesis that the curvature is pinched toward positive.

Lemma 5.6. There exists α ′ ∈ (0, α] such that Condition (NC) κ/16 holds for {g(t)} t∈[T A ,bmax+α ′ ) .

Proof. Let x ∈ M(b max ), t ∈ (b max , b max + α ′ ) and ρ ∈ (0, 10 -3 ) be such that | Rm | ≤ ρ -2 on P (x, t, ρ, -ρ 2 ). Choosing α ′ small enough, we can ensure that B(x, b max , ρ/2) ⊂ B(x, t, ρ) and moreover that

P (x, b max , ρ/2, -ρ 2 /4) ⊂ P (x, t, ρ, -ρ 2 ). It follows that | Rm | ≤ 4ρ -2 on P (x, b max , ρ/2, -ρ 2 /4). Since (CN) κ is satisfied up to time b max , we deduce that volB(x, b max , ρ/2) ≥ κ(ρ/2) 3 . Again by proper choice of α ′ , vol g(t) B(x, b max , ρ/2) is at least half of volB(x, b max , ρ/2). Hence volB(x, t, ρ) ≥ vol g(t) B(x, b max , ρ/2) ≥ 1 2 volB(x, b max , ρ/2) ≥ κ 16 ρ 3 .
Applying Proposition B, we deduce that {(M(t), g(t))} t∈[T A ,bmax+α ′ ) is an (r, δ)-surgical solution. By Proposition C, it is an (r, δ, κ)-surgical solution. This contradicts maximality of b max .

Case 2 R max (b max ) = Θ.
Proposition A yields a riemannian manifold (M + , g + ). If M + is empty, then the construction stops. Suppose M + = ∅. Applying Shi's short time existence theorem for Ricci flow on M + with initial metric g + , we obtain a positive number α and an evolving metric {g(t)} t∈(bmax,bmax+α) on M + whose limit from the right as t tends to b max is equal to g + . Since R max (g + ) ≤ Θ/2, we may also assume that R max remains bounded above by Θ. By Proposition 3.1 it has curvature pinched toward positive. Setting M(t) := M + for t ∈ (b max , b max +α), we obtain an evolving manifold {(M(t), g(t))} t∈[T A ,bmax+α) satisfying the first two properties of the definition of (r, δ)-surgical solutions.

Lemma 5.7. There exists α ′ ∈ (0, α] such that Condition (NC) κ/16 holds on

[T A , b max + α ′ ). Proof. Let x ∈ M(b max ), t ∈ (b max , b max + α ′ ) and ρ ∈ (0, 10 -1 ) be such that | Rm | ≤ ρ -2 on P (x, t, ρ, -ρ 2 ). If B(x, t, ρ/2
) is unscathed and stays so until b max , then we can repeat the argument used to prove Lemma 5.6. Otherwise it follows from the assumption κ ≤ κ st and properties of almost standard caps.

Applying Proposition B, we deduce that {(M(t), g(t))} t∈[T A ,bmax+α ′ ) is an (r, δ)-surgical solution. By Proposition C, it is an (r, δ, κ)-surgical solution. Again this contradicts the assumption that b max should be maximal.

Choosing cutoff parameters

In this section, we give some technical results necessary to prove Theorem 5.1. Their statements are nearly identical to those of the corresponding results of [BBB + , Section 4], surgical solutions replacing Ricci flow with bubbling-off. The proofs are also almost identical, the minor adaptations being precised below.

Bounded curvature at bounded distance

We have the following technical lemmas, as in [BBB + , Section 4.2]: Lemma 6.1 (Local curvature-distance lemma). Let (U, g) be a Riemannian manifold. Let Q 1, C > 0, x ∈ U and set Q x =| R(x) | +Q. Suppose that there exist y ∈ U such that R(y) 2Q x , and a minimising segment [

holds at (x, s) if R(x, s) Q. Then t 2 -t (2CQ x ) -1 . Lemma 6.3 (Local curvature-control lemma). Let Q > 0, C > 0, ε ∈ (0, 2ε 0 ],
and {(M(t), g(t))} t∈I be a surgical solution on M. Let (x 0 , t 0 ) ∈ M ×I and set Q 0 = |R(x 0 , t 0 )|+Q. Suppose that

P = P (x 0 , t 0 , 1 2C √ Q 0 , -1 8CQ 0
) is unscathed and that each (x, t) ∈ P with R(x, t) Q has an (ε, C)-canonical neighbourhood. Then for all (x, t) ∈ P , R(x, t) 4Q 0 .

We shall use repeatedly the following well-known consequences of curvature pinching: Proposition 6.4. Let (U k , g k (•), * k ) be a sequence of pointed evolving metrics defined on intervals I k ⊂ R + , and having curvature pinched toward positive. Let (x k , t k ) ∈ U k × I k be a sequence such that (1 + t k )R(x k , t k ) goes to +∞. Then the sequence ḡk := R(x k , t k )g(t k ) has the following properties: i. The sequence R min (ḡ k ) tends to 0.

ii. If (U k , ḡk , * k ) converges in the pointed C 2 sense, then the limit has nonnegative curvature operator.

We also recall: Lemma 6.5. Let ε ∈ (0, 10 -1 ]. Let (M, g) be a riemannian 3-manifold, N ⊂ M be an ε-neck, and S be a middle sphere of N. Let [xy] be a geodesic segment such that x, y ∈ M \ N and [xy] ∩ S = ∅. Then the intersection number of [x, y] with S is odd. In particular, if S is separating in M, then x, y lie in different components of M \ S.

Proof. [BBB + , Lemma 1.

3.2.]

We summarise the conclusion of Lemma 6.5 by saying that N is traversed by the segment [xy]. Corollary 6.6. Let ε ∈ (0, 10 -1 ]. Let (M, g) be a riemannian 3-manifold, U ⊂ M be an ε-cap and V be a core of U. Let x, y be points of M \ U and [xy] a geodesic segment connecting x to y.

Then [xy] ∩ V = ∅.
As for Ricci flow with bubbling-off, we then have Theorem 6.7 (Curvature-distance). For all A, C > 0 and all ε ∈ (0, 2ε 0 ], there exist Q = Q(A, ε, C) > 0 and Λ = Λ(A, ε, C) > 0 with the following property. Let I ⊂ [0, +∞) be an interval and {(M(t), g(t))} t∈I be a surgical solution with curvature pinched toward positive. Let (x 0 , t 0 ) ∈ M be such that:

i. R(x 0 , t 0 ) ≥ Q; ii. For each point y ∈ B(x 0 , t 0 , AR(x 0 , t 0 ) -1/2 ), if R(y, t) ≥ 2R(x 0 , t), then (y, t) has an (ε, C)-canonical neighbourhood.
Then for all y ∈ B(x 0 , t 0 , AR(x 0 , t 0 ) -1/2 ), we have R(y, t 0 ) R(x 0 , t 0 ) ≤ Λ.

Proof. It suffices to redo the proof of [BBB + , Theorem 4.2.1], with the following minor differences:

• In Step 1, to control the injectivity radius, one can use property iii) in the definition of (ε, C)-canonical neighbourhoods, as the canonical neighbourhood considered is not ε-round.

• In Step 2, to prove that [x ′ k y ′ k ] is covered by strong ε-necks, one has to rule out closed canonical neighbourhoods. This is clear by the curvature ratio properties. Then use Corollary 6.6 instead of [BBB + , Lemma 1.3.2]

Existence of cutoff parameters

For the convenience of the reader, we restate Theorem 5.1. Theorem 6.8 (Cutoff parameters). For all r, δ > 0, there exist h ∈ (0, δr) and D > 10 such that if (M(•), g(•)) is a complete surgical solution of bounded curvature defined on an interval [a, b], with curvature pinched toward positive and satisfying (CN) r , then the following holds:

Let t ∈ [a, b] and x, y, z ∈ M(t) such that R(x, t) ≤ 2/r 2 , R(y, t) = h -2 , and R(z, t) ≥ D/h 2 . Assume there is a curve γ connecting x to z and containing y, such that each point of γ with scalar curvature in

[2C 0 r -2 , C 0 -1 Dh -2 ] is centre of a ε 0 -neck. Then (y, t) is centre of a strong δ-neck.
Proof. The proof is almost the same as for [BBB + , Theorem 4.3.1], arguing by contradiction. We only need to adapt Step 1.

Fix two constants r > 0, δ > 0, sequences h k → 0, D k → +∞, a sequence (M k (•), g k (•)) of surgical solutions satisfying the above hypotheses, and sequences

t k > 0, x k , y k , z k ∈ M such that R(x k , t k ) ≤ 2r -2 , R(z k , t k ) ≥ D k h -2 k , and R(y k , t k ) = h -2 k . Let γ k be a curve from x k to z k such that y k ∈ γ k , whose points of scalar curvature in [2C 0 r -2 , C 0 -1 Dh -2 ] are centre of ε 0 -neck. Fi- nally assume that (y k , t k ) is not centre of a strong δ-neck.
Consider the sequence ( Mk (•), ḡk (•)) defined by the following parabolic rescaling ḡk (t) = h -2 k g k (t k + th 2 k ) . In order to clarify notation, we shall put a bar on points when they are involved in geometric quantities computed with respect of the metric ḡk . Thus for instance, we have R(ȳ k , 0) = 1. The contradiction will come from extracting a converging subsequence of the pointed sequence ( Mk (•), ḡk (•), ȳk , 0) and showing that the limit is the cylindrical flow on S 2 × R, which implies that for k large enough, y k is centre of some strong δ-neck, contradicting our hypothesis.

Step 1. ( Mk (0), ḡk (0), ȳk ) subconverges in the pointed C ∞ sense to (S 2 × R, g ∞ ) where g ∞ is a product metric of nonnegative curvature operator and scalar curvature at most 2.

Proof. First we control the curvature on balls around ȳk . Since R(y k , t k ) goes to +∞, Theorem 6.7 implies that for all ρ > 0, there exists Λ(ρ) > 0 and k 0 (ρ) > 0 such that ḡk (0) has scalar curvature bounded above by Λ(ρ) on B(y k , ρ) for k ≥ k 0 (ρ). Moreover, by Assumption (iii) of the definition of canonical neighbourhoods, ḡk (

•) is C 0 -1 -noncollapsed at (y k , 0). Indeed R(y k , t k ) = h k -2 ∈ [2C 0 r -2 , C 0 -1 Dh -2 ], hence y k is centre of a ε 0 -neck.
Thus we can apply Gromov's compactness theorem to extract a converging subsequence with regularity C 1,α . Let us prove that for large k, the ball B(ȳ k , ρ) is covered by ε 0 -necks. Recall that g k (t k ) satisfies

|∇R| < C 0 R 3/2 ,
at points covered by canonical neighbourhoods. Take a point y such that R(y, t k ) ≤ 2C 0 r -2 and integrate the previous inequality on the portion of [y k y] where R ≥ 2C 0 r -2 . An easy computation yields

d(ȳ, ȳk ) ≥ 1 h k 2 C 0 ( r √ 2C 0 -h k ) ρ, , (9)
for k larger than some k 1 (ρ) ≥ k 0 (ρ). It follows that the scalar curvature of g k (t k ) is at least 2C 0 r -2 on B(ȳ k , ρ) for every integer k ≥ k 1 (ρ). It follows that x k / ∈ B(ȳ k , ρ) and that B(ȳ k , ρ) is covered by (ε 0 , C 0 )-canonical neighbourhoods. On the other hand, for k larger than some k 2 (ρ), we have

R(ȳ, 0) Λ(ρ) < C 0 -1 Dh -2 for all ȳ ∈ B(ȳ k , ρ). It follows that γ ∩ B(ȳ k , ρ) is covered by ε 0 -necks. As z k / ∈ B(ȳ k , ρ), it follows that B(ȳ k , ρ
) is contained in the union U ρ,k of these necks: indeed, every segment coming from ȳk and of length less than ρ lies there. Now by the (CN) r assumption, these necks are strong ε 0 -necks. The scalar curvature on B(ȳ k , ρ) being less than Λ(ρ) for k k 0 , we deduce that on each strong neck, ḡk (t) is smoothly defined on [-1 2Λ(ρ) , 0], and has curvature bounded above by 2Λ(ρ). Hence for each ρ > 0, the parabolic balls P (ȳ k , 0, ρ, -1 2Λ(ρ) ) are unscathed, with scalar curvature bounded above by 2Λ(ρ) for all k ≥ k 2 (ρ). Since g k (•) has curvature pinched toward positive, this implies a uniform control of the curvature operator there.

Hence we can apply the local compactness theorem B.1. Up to extracting, ( Mk (0), ḡk (0), ȳk ) converges to some complete noncompact pointed riemannian 3-manifold (M ∞ , ḡ∞ , ȳ∞ ). By Proposition 6.4, the limit has nonnegative curvature operator.

Passing to the limit, we get a covering of M ∞ by 2ε 0 -necks. Then Proposition 7.5 shows that M ∞ is diffeomorphic to S 2 × R. In particular, it has two ends, so it contains a line, and Toponogov's theorem implies that it is the metric product of some (possibly nonround) metric on S 2 with R.

As a consequence, the spherical factor of this product must be 2ε 0 -close to the round metric on S 2 with scalar curvature 1. Hence the scalar curvature is bounded above by 2 everywhere. This finishes the proof of Step 1.

Henceforth we pass to a subsequence, so that ( Mk (0), ḡk (0)) satisfies the conclusion of Step 1.

The rest of the proof is the same as for [BBB + , Theorem 4.3.1].

Proof of Proposition A 7.1 Piecing together necks and caps

Definition. An ε-tube is an open subset U ⊂ M which is equal to a union of ε-necks, and whose closure in M is diffeomorphic to S 2 × I, S 2 × R, or S 2 × [0, +∞).

Proposition 7.1. Let ε ∈ (0, 2ε 0 ]. Let (M, g) be a connected, orientable riemannian 3-manifold. Let X be a closed, connected subset of M such that every point of X is the centre of an ε-neck or an ε-cap. Then there exists an open subset U ⊂ M containing X such that either i. U is equal to M and diffeomorphic to

S 3 , S 2 × S 1 , RP 3 , RP 3 #RP 3 , R 3 , S 2 × R, or a punctured RP 3 , or ii. U is a 10ε-cap, or iii. U is a 10ε-tube.
Proof. First we deal with the case where X is covered by ε-necks.

Lemma 7.2. If every point of X is centre of an ε-neck, then there exists an open set U containing X such that U is a 10ε-tube, or U is equal to M and diffeomorphic to S 2 × S 1 or S 2 × R.

Proof. Let x 0 be a point of X and N 0 be a 10ε-neck centred at x 0 , contained in an ε-neck U 0 , also centred at x 0 . If X ⊂ N 0 we are done. Otherwise, since X is connected, we can pick a point x 1 ∈ X ∩ N 0 and a 10ε-neck N 1 centred at x 1 , with x 1 arbitrarily near the boundary of N 0 . By Lemma 4.4, an appropriate choice of x 1 ensures that N 1 ⊂ U 0 and the middle spheres of N 0 and N 1 are isotopic. In particular, the closure of

N 0 ∪ N 1 is diffeomorphic to S 2 × I, so N 0 ∪ N 1 is a 10ε-tube.
If X ⊂ N 0 ∪ N 1 then we can stop. Otherwise, we pick a 10ε-neck N 2 centred at some point x 2 near the boundary component of N 1 that does not lie in N 0 , and iterate the construction as long as possible. Three cases may occur.

Case 1 The construction stops with some 10ε-tube N 0 ∪• • •∪N k containing X. Then we are done.

Case 2 The construction stops with some 10ε-tube N 0 ∪ • • • ∪ N k such that adding another neck N k+1 does not produce a 10ε-tube.

This can only happen if N k+1 ∩N 0 is non empty. In this case, by adjusting the centre x k+1 of N k+1 , we can ensure that N 0 , . . . , N k+1 cover M and that the intersection of N k+1 and N 0 is topologically standard. In this case, M fibers over the circle with fiber S 2 . Since M is orientable, it follows that M is diffeomorphic to S 2 × S 1 .

Case 3 The construction can be iterated forever.

In this case, the union U of all N k 's is a 10ε-tube.

Claim. The frontier of U is connected, equal to the boundary component of N0 which does not lie in N 1 .

We prove the claim by contradiction. If it is not true, then we can pick two points x, y ∈ X, each one being close to a distinct component of the frontier of U. Since U ∩ X is connected, we can find a path γ connecting x to y in U ∩ X. Now γ is compact, so it can be covered by finitely many 10ε-necks, each of which is contained in some ε-neck. We thus obtain a finite collection of ε-necks which cover U. Hence U is relatively compact. This shows that the scalar curvature is bounded on U. Hence each N k has a definite size, and adding each N k to N 0 , . . . , N k-1 adds definite volume. It follows that volU is infinite, which is a contradiction. This proves the claim.

We continue the proof of Lemma 7.2. If U contains X, then we are done. Otherwise, we pick a point x -1 ∈ N 0 ∩ X close to the frontier, and choose a neck N -1 centred at x -1 . We perform the same iterated construction as before. At each stage, we have a 10ε-tube N -k ∪ • • • ∪ N -1 ∪ U whose frontier is connected. Hence the analogue of Case 2 above cannot occur. If the construction stops, then we have found a 10ε-tube containing X. Otherwise the union V of all N k 's for k ∈ Z is a 10ε-tube. Repeating the argument used to prove the claim, we see that the frontier of V is empty. Since M is connected, it follows that

V = M ∼ = S 2 × R.
To complete the proof of Proposition 7.1, we need to deal with the case where there is a point x 0 ∈ X which is the centre of an ε-cap C 0 . By definition of a cap, some collar neighbourhood of the boundary of C 0 is an ε-neck U 0 . If X ⊂ C 0 , pick a point x 1 close to the boundary of C 0 . If x 1 is centre of a 10ε-neck N 1 , then we apply Lemma 4.4 again to find that C 1 := C 0 ∪ N 1 is a 10ε-cap. Again we iterate this construction until one of the following things occur:

Case 1 The construction stops with a 10ε-cap containing X.

Case 2 The construction stops with a 10ε-cap

C k = C 0 ∪ • • • ∪ N k and a point x k+1 near its frontier, such that x k+1 is centre of a 10ε-cap C whose boundary is contained in C k . Then C k ∪ C equals M and is diffeomorphic to S 3 , RP 3 , or RP 3 #RP 3 .
Case 3 The construction goes on forever. Then the same volume argument as in the proof of the above Claim shows that the union of all C ′ k s is M. Thus M itself is a 10ε-cap, diffeomorphic to R 3 or a punctured RP 3 .

Putting X = M, we obtain the following corollary:

Theorem 7.3. Let ε ∈ (0, 2ε 0 ]. Let (M, g) be a connected, orientable rie- mannian 3-manifold. If every point of M is the centre of an ε-neck or an ε-cap, then M is diffeomorphic to S 3 , S 2 ×S 1 , RP 3 , RP 3 #RP 3 , R 3 , S 2 ×R, or a punctured RP 3 .
Here is another consequence of Proposition 7.1: Corollary 7.4. Let ε ∈ (0, 2ε 0 ]. Let (M, g) be an orientable riemannian 3-manifold. Let X be a closed submanifold of M such that every point of X is the centre of an ε-neck or an ε-cap. Then one of the following conclusions holds:

i. M is diffeomorphic to S 3 , S 2 × S 1 , RP 3 , RP 3 #RP 3 , R 3 , S 2 × R, or a punctured RP 3 , or
ii. There exists a locally finite collection N 1 , . . . , N p of 10ε-caps and 10εtubes with disjoint closures such that X ⊂ i N i .

Proof. We apply Proposition 7.1 to each connected component of X. If Case (i) of the required conclusion does not hold, then we have found a locally finite collection of 10ε-caps and 10ε-tubes which cover X. By merging some of them if necessary, we can ensure that they have disjoint closures.

Finally, we have a more precise result when there are just necks:

Proposition 7.5. Let ε ∈ (0, 2ε 0 ]. Let (M, g) be an open riemannian 3- manifold such that every point of M is centre of an ε-neck. Then M is diffeomorphic to S 2 × R.
This follows immediately from the proof of Lemma 7.2.

In order to apply Theorem 5.1, one has to prove the following

Claim. Each point of γ with scalar curvature in [2C 0 r -2 , C 0 -1 Dh -2 ] is centre of some ε 0 -neck.
Proof of the claim. Let y ∈ γ be such a point. By the curvature assumptions, y is centre of a (ε 0 , C 0 )-canonical neighbourhood U, disjoint from x and z. Hence U cannot be a closed manifold. It remains to rule out the (ε 0 , C 0 )-cap case. We argue by contradiction. Assume that U is an (ε 0 , C 0 )-cap. Then

U = N ∪ C, where N is a ε 0 -neck, N ∩ C = ∅, N ∩ C = ∂C and y ∈ IntC.
For simplicity dilate the metric by a factor such that the scalar curvature of N is close to 1. Denote by S the middle sphere of N. The curve γ is clearly not minimizing in U. In particular if x ′ (resp. z ′ ) is an intersection point of γ with S lying between x and y (resp. y and z),

then d(x ′ , z ′ ) ≤ diam(S) << 2ε 0 -1 < d(x ′ , y) + d(y, z ′ ). The geodesic segment [x ′ z ′ ] ⊂ U is not contained in X,
otherwise this would contradict the minimality of γ in X. Hence there exists p ∈ [x ′ , z ′ ] ∩ ∂X. By definition of X, the corresponding component of ∂X is a boundary component of some neck N i . Let us prove that γ intersects N i , which is a contradiction. Denote by S i + the above boundary component of N i , and note that d(S i + , S) < diam(S). Let N ′ be a 10ε 0 -subneck of N i which admits S i + as a boundary component, and p ′ ∈ N ′ be its centre. Then d(p ′ , S) < diam(S) + (10ε 0 ) -1 < (4ε 0 ) -1 . It follows from Lemma 4.4 that S ′ is isotopic to S in N. In particular γ intersects S ′ .

Let y be a point of γ of scalar curvature h -2 . Theorem 5.1 yields a cutoff neck N centred at y. Any δ-neck meeting N has to be traversed by γ, so N is disjoint from the N i 's. This contradicts maximality of {N i }.

Having established Lemma 7.6, we prove Proposition A. Let {N i } be a collection of cutoff necks given by that lemma. Applying Theorem 3.4, we obtain a Riemannian manifold (M ′ , g + ). By construction, the components of M ′ fall into two types. Either they have curvature less than Θ/2, or they are covered by canonical neighbourhoods. Applying Theorem 7.3, we may safely throw away the components of the second type, obtaining the manifold (M + , g + ). We remark that the operation cannot decrease R min (in fact R min (g + ) is equal to R min (g(b)) unless M + is empty, in which case it is equal to +∞). Thus the proof of Proposition A is complete.

t → (M + (a), g + (a)) if t = a (M(t), g(t)) if t ∈ (a, b]
We shall still denote by g(•) the restriction. Given (x, t) ∈ M, r > 0 and ∆t > 0 we define the forward parabolic neighbourhood P (x, t, r, ∆t) as the set

P (x, t, r, ∆t) = {(x ′ , t ′ ) ∈ M | x ′ ∈ B(x, t, r), t t ′ t + ∆t}
When we consider a restriction of g(•) to some [a, b] ⊂ I, the parabolic neighbourhood P (x, a, r, ∆t) will be defined using the ball B(x, a, r) of radius r with respect to the metric g + (a). A parabolic neighbourhood P (x, t, r, ∆t) is said to be unscathed if x ′ ∈ M reg (t ′ ) for all x ′ ∈ B(x, t, r) and t ′ ∈ [t, t + ∆t). Otherwise it is scathed. Given two surgical solutions (M(•), , g(•)) and (M 0 (•), g 0 (•)), we say that an unscathed parabolic neighbourhood P (x, t, r, ∆t) of (M(•), g(•)) is ε-close to another unscathed parabolic neighbourhood P (x 0 , t, r 0 , ∆t) of (M 0 (•),

g 0 (•)) if (B(x, t, r), g(•)) is ε-close to (B(x 0 , t, r 0 ), g 0 (•)) on [t, t + ∆t] . We say that P (x, t, r, ∆t) is ε-homothetic to P (x 0 , t 0 , r 0 , λ∆t)) if it is ε-close after a parabolic rescaling by λ.
The goal of this section is to prove the following technical theorem:

Theorem 8.1 (Persistence of almost standard caps). For all A > 0, θ ∈ [0, 1) and r > 0, there exists δ = δper (A, θ, r) with the following property. Let (M(•), g(•)) be a surgical solution defined on some interval [a, b], which is a (r, δ)-surgical solution on [a, b), with r r and δ δ. Let t 0 ∈ [a, b) be a singular time and consider the restriction of (M(•), g(•)) to [t 0 , b]. Let p ∈ (M(t 0 ), g(t 0 )) be the tip of some δ-almost standard cap of scale h. Let t 1 ≤ min(b, t 0 + θh 2 ) be maximal such that P (p, t 0 , Ah, t 1t 0 ) is unscathed. Then the following holds: i. The parabolic neighbourhood P (p, t 0 , Ah, t

1 -t 0 ) is A -1 -homothetic to P (p 0 , 0, A, (t 1 -t 0 )h -2 ); ii. If t 1 < min(b, t 0 + θh 2 ), then B(p, t 0 , Ah) ⊂ M sing (t 1 ) disappear at time t 1 .
Remark. In [BBB + ], the conclusion in the last case was that B(p, t 0 , Ah) ⊂ M sing (t 1 ).

Before giving the proof of the theorem, we summarize some technical results from [BBB + ] Sections 6.1 and 6.2. Let T 0 be a positive real number. Let M 0 = (X 0 , g 0 (•), p 0 ) (the model) be a complete 3-dimensional Ricci flow defined on [0, T 0 ] such that the quantity Λ(N) := max

X 0 ×[0,T 0 ] |∇ p Rm | | 0 ≤ p ≤ N is finite for all N ∈ N.
Corollary 8.2 (Persistence of the model in dimension 3). Let A > 0, there exists ρ = ρ(M 0 , A) > A with the following property. Let {(M(t), g(t))} t∈[0,T ] be a surgical solution with T ≤ T 0 . Suppose that

a) (M(•), g(•)) has curvature pinched toward positive. b) | ∂R ∂t | ≤ C 0 R 2 at any (x, t) with R(x, t) 1. Let p ∈ M(0) and t ∈ (0, T ] be such that c) B(p, 0, ρ) is ρ -1 -close to B(p 0 , 0, ρ) ⊂ X 0 , d) P (p, 0, ρ, t) is unscathed and | Rm | Λ([A] + 1) there. Then P (p, 0, A, t) is A -1 -close to P (p 0 , 0, A, t).
Proof. The proofs of [BBB + , Corollary 6.2.4 and 6.2.6] work for surgical solutions.

Proof of Theorem 8.1. We consider as model the standard solution X 0 := (S 0 , g 0 (•), p 0 ) restricted to [0, θ]. Let us assume for simplicity that T ≥ t 0 + θh 2 , so that t 1 = t 0 + θh 2 . For any nonnegative integer N, recall that

Λ(N) = max S 0 ×[0,θ] |∇ p Rm |, |R|; 0 ≤ p ≤ N .
In the sequel we consider the restriction of (M(•), g(•)) to [t 0 , b] and we define:

ḡ(t) := h -2 g(t 0 + th 2 ) for t ∈ [0, min{θ, (b -t 0 )h -2 }].
Note that ḡ(•) satisfies Assumptions a) and b) of Corollary 8.2. Indeed, it is readily checked that the curvature pinched toward positive property is preserved by the parabolic rescaling, since t 0 0 and h -2 1. On the other hand, if g(•) satisfies (CN) r on [0, b), it follows easily by a continuity argument that any (x, b) with R(x, b) 2r(b) -2 satisfies the estimate ∂R ∂t C 0 R 2 at (x, b). After rescaling by h(b) -2 >> 2r -2 , this property holds at points with scalar curvature above 1.

Fix A > 0 and set ρ := ρ(M 0 , A). By the definition of an δ-almost standard cap, the ball B ḡ(p, 0, 5 + δ -1 ) is δ ′ -close to B(p 0 , 0, 5 + δ -1 ) ⊂ S 0 . Let T max ∈ [0, θ] be the maximal time such that P ḡ(p, 0, A, T max ) is unscathed. By closeness at time 0 one has |R ḡ| ≤ 2Λ 0 on B ḡ(p, 0, δ -1 ). Now for t ∈ [0, min((4ΛC 0 ) -1 , θ)] such that P ḡ(p, 0, ρ, t) is unscathed, we have |R ḡ| ≤ 4Λ 0 on P ḡ(p, 0, ρ, t) by the time derivative estimate on the scalar curvature. Using the pinching assumptions, we deduce | Rm ḡ | ≤ Λ ′ 0 on the same neighbourhood.

Set T ×2 := min(θ, (4Λ 0 C 0 ) -1 , (4Λ ′ 0 ) -1 ). The above curvature bound gives, for t ≤ min(θ, T ×2 ), 1/2 ≤ ḡ(t)/ḡ(0) ≤ 2 on B ḡ(p, 0, δ -1 ). In particular, for all x ∈ B ḡ(p, 0, ρ) and all 0 < t ≤ min(θ, T ×2 ), (x, t) is not centre of a δ-neck because d ḡ(t) (x, p) ≤ 2d ḡ(0) (x, p) ≤ 2ρ and the length of a δ-neck at time t is larger than 1

/2δ -1 R(x, t) -1/2 ≥ (4δΛ 0 ) -1 > 4ρ, if δ is small enough. This implies that P ḡ(p, 0, ρ, t) is unscathed if t ≤ min(T ×2 , T max ). Indeed, if not, there exists t ′ < t such that P ḡ(p, 0, ρ, t ′ ) is unscathed but B ḡ(p, 0, ρ) ∩ M sing (t ′ ) = ∅. If B ḡ (p, 0, ρ) is not contained in M sing (t ′
), then it must intersect a surgery sphere of S(t ′ ), which is the middle sphere of a δ-neck centred at (x, t ′ ). The above estimate rules out this possibility. Hence B ḡ(p, 0, A) ⊂ B ḡ (p, 0, ρ) ⊂ M sing (t ′ ) for t ′ < T max . This is impossible by assumption. Remark that if B(p, 0, ρ) ∩ M sing (t) = ∅ the same arguments shows that B(p, 0, ρ) is contained in M sing (t) and disappears at time t.

We can now apply Corollary 8.2. We get that P ḡ(p, 0, ρ 1 , t) is ρ -1 1 -close to P (p 0 , 0, ρ 1 , t) for all t ≤ min(T ×2 , T max ). If T ×2 ≥ T max we are done.

Otherwise by closeness we have that | Rm ḡ | and |R ḡ| are no greater than 2Λ 0 on P ḡ(p, 0, ρ 1 , T ×2 ). Then |R ḡ| ≤ 4Λ 0 on P ḡ(p, 0, ρ 1 , min(2T ×2 , T max )) where ρ 1 = ρ(M 0 , ρ 2 ) and ρ 2 > 0. We then iterate the above argument, which terminates in finitely many steps, as in [BBB + , Corollary 6.2.4]. If T max < θ, then Conclusion (ii) follows from the previous remark. This finishes the proof of Theorem 8.1.

Proposition B

Recall the statement of Proposition B: Proposition B. For all Q 0 , ρ 0 , κ > 0 there exist r = r(Q 0 , ρ 0 , κ) < 10 -3 and δB = δB (Q 0 , ρ 0 , κ) > 0 with the following property: let δ ≤ δB , 0 ≤ T A < b and (M(•), g(•)) be a surgical solution defined on [T A , b] such that g(T A ) satisfies | Rm | ≤ Q 0 and has injectivity radius at least ρ 0 .

Assume that (M(•), g(•)) satisfies Condition (NC) κ/16 , has curvature pinched toward positive, and that for each singular time t 0 , (M + (t 0 ), g + (t 0 )) is obtained from (M(•), g(•)) by (r, δ)-surgery at time t 0 .

Then (M(•), g(•)) satisfies Condition (CN) r .

We recall a lemma from [BBB + ].

Lemma 9.1 (distance distorsion). Let {g(t)} be a Ricci flow solution on a n-dimensional manifold U, defined for t

∈ [t 1 , t 2 ]. Suppose that | Rm |≤ Λ on U × [t 1 , t 2 ]. Then e -2(n-1)Λ(t 2 -t 1 ) ≤ g(t 2 ) g(t 1 ) ≤ e 2(n-1)Λ(t 2 -t 1 )
Proof. [BBB + , Lemma 0.6.6.].

In order to prove Proposition B, we argue by contradiction. Suppose that some fixed numbers Q 0 , ρ 0 , κ > 0 have the property that for all r ∈ (0, 10 -3 ) and δB > 0 there exist counterexamples. Then we can consider sequences r k → 0, δ k → 0, and a sequence of (r k , δ k , κ)-surgical solutions (M k (•), g k (•)) on [0, b) which satisfy Condition (NC) κ/16 , have curvature pinched toward positive, and such that for each singular time t 0 , (M k,+ (t 0 ), g k,+ (t 0 )) is obtained from (M(•), g(•)) by (r, δ)-surgery at time t 0 , but (CN) r k fails for some t k . This last assertion means that there exists x k ∈ M k (t k ) such that

Q k := R(x k , t k ) ≥ r -2 k ,
and yet (x k , t k ) does not have a (ε 0 , C 0 )-canonical neighbourhood. By a standard point-picking argument (see [KL08, Lemma 52.5]), we may choose the sequence of bad points (x k , t k ) and

H k → +∞ such that for all t ∈ [t k -H k Q -1 k , t k ] and x ∈ M k (t), if R(x, t) ≥ 2Q k then (x, t) has a (ε 0 , C 0 )- canonical neighbourhood.
Without loss of generality, we assume that

δ k ≤ δ(k, 1 - 1 k , r k )
(the right-hand side being the parameter given by the Persistence Theorem 8.1). From now on, the proof follows the lines of [BBB + , Section 7.2]. We need a preliminary lemma.

Lemma 9.2 (Parabolic balls of bounded curvature are unscathed). For all K > 0, ρ > 0 and τ > 0 there exists an integer

k 0 = k 0 (K, ρ, τ ) such that for all k k 0 , if | Rm | K on B ḡk (x k , 0, ρ) × (-τ, 0] then P ḡk (x k , 0, ρ, -τ ) is unscathed.
Proof. Arguing by contradiction, fix k and assume that there exist z k ∈ B ḡk (x k , 0, ρ) and s k ∈ [-τ, 0) such that z k / ∈ M reg (s k ). As z k exists after s k , there is an added cap V in M + (s k ) such that z k ∈ V . We can take s k to be maximal satisfying this property, i.e. the set B ḡk (x k , 0, ρ) × (s k , 0] is unscathed. In the sequel, we consider the restriction of ḡk (•) to [s k , 0], as explained in the beginning of Section 8. We drop indices for simplicity.

By definition, there exists a marked δ-almost standard cap (U, V, p, y) such that (U, R ḡ(y, s)ḡ(y, s)) is δ ′ -close to B(p 0 , 5 + δ -1 ) ⊂ S 0 . Set g(y, s) := R ḡ (y, s)ḡ(y, s)). We shall now show that d g(x, p) is bounded independently of k (if k is large enough.) Notice that since y ∈ ∂V , R ḡ(y, s) = R ḡ(y, s). We now get

V W ≤ 5 δ -1
d g(x, y) ≤ √ 12Ke 2Kτ =: K ′ (K, ρ, τ ).
Finally, d g(x, p) ≤ K ′ + 5 .

Let A > 2(K ′ + 5) and θ < 1. The persistence theorem 8.1 implies that the set P ḡ(p, s, AR ḡ(y, s) -1/2 , min(θR ḡ(y, s) -1 , |s|) is, after parabolic rescaling at time s, A -1 -close to P (p 0 , 0, A, min(θ, |s| Rk (y, s))) . Indeed, the second alternative of the persistence theorem does not occur since in this case, x ∈ B g(p, A) ⊂ M sing (t + ) for some t + ∈ (s, 0), and x disappears at time t + .

We now choose A := k and θ := 1 -1/k.

Claim. We have s + θR ḡ (y, s) -1 > 0 for large k.

Case 2 The point (x ′ , s ′ ) has no (ε, C st )-canonical neighbourhood. Then s ′ ≤ 3/4 and x ′ / ∈ B(p 0 , 0, ǫ -1 st ). Hence we have

d g(x, s ′ , p) 9/10ε -1 st 3/2(εβ) -1 > (εβ) -1 + 5.
We infer that (x, -s ′ ) is centre of an (ε 0 β)-neck, coming from the strong δ-neck there at the singular time. We now apply the neck strengthening lemma 4.5 which asserts that (x, 0) is centre of a strong ε-neck. Indeed the closeness with the standard solution ensures that P ḡ(x, -s ′ , (ε 0 β) -1 , 0) ⊂ P ḡ(x, -s ′ , A, 0) is unscathed and has | Rm | 2K st . This proves Lemma 9.2. Now we begin the proof of Proposition B proper. We consider parabolic rescalings.

Step 1. The sequence ( Mk (0), ḡk (0), xk ) subconverges to some complete pointed riemannian manifold (M ∞ , g ∞ , x ∞ ) of nonnegative curvature operator.

Proof. We have to show that the sequence satisfies the hypothesis of the local compactness theorem for flows (Theorem B.1.) By choice of the basepoint and curvature pinching, we can apply Theorem 6.7. Hence for every ρ, the scalar curvature of ḡk (0) is bounded above on B(x k , 0, ρ) by some constant Λ(ρ) if k ≥ k(ρ).

Next we wish to obtain similar bounds on parabolic balls P (x k , 0, ρ, -τ (ρ)) for some τ (ρ) > 0, and show that they are unscathed. Set C(ρ) := Λ(ρ) + 2. Let k 1 (ρ) := k 0 (K(ρ), ρ, (2C 0 C(ρ) -1 ) be the parameter given by Lemma 9.2.

Claim. If k ≥ k 1 (ρ), then P ḡk (x k , 0, ρ, -(2C 0 C(ρ)) -1 ) is unscathed and sat- isfies | Rm | 2C(ρ). Proof. Choose s = s k ∈ [-(4C 0 C(ρ)) -1
, 0] minimal such that Bk (x k , 0, ρ) × (s, 0] is unscathed. By the curvature-time lemma 6.2 we have R 2C(ρ) on this set, which implies | Rm | 2C(ρ) by the Pinching Lemma 3.3. By Lemma 9.2, P ḡk (x k , 0, ρ, s) is unscathed. By minimality of s we then have

s = -(2C 0 C(ρ)) -1 .
By hypothesis, the solutions g k (•) are κ-noncollapsed on scales less than r 0 . Hence ḡk (0) is κ-noncollapsed on scales less than √ Q k r 0 . This, together with the curvature bound implies a positive lower bound for the injectivity radius at (x k , 0). Hence Theorem B.1 applies to the sequence ( Mk , ḡk (•), x k ). It implies that the sequence subconverges to (M ∞ , g ∞ (•), x ∞ ), where M ∞ is a smooth manifold, g ∞ (0) is complete and g ∞ (.) is defined on B(x ∞ , 0, ρ) × (-(2C 0 C(ρ)) -1 , 0] for each ρ > 0.

Lastly, since the metrics g k (0) are normalised and the scaling factor Q k goes to +∞, the limit metric g ∞ (t) has nonnegative curvature operator by curvature pinching. This argument completes the proof of Step 1.

Since R(x ∞ ) = 1, M ∞ is a complete, nonflat, nonnegatively curved riemannian 3-manifold. By the Cheeger-Gromoll-Hamilton classification of such manifolds, M is diffeomorphic to R 3 , S 1 × R 2 , S 1 × S 2 , a line bundle over a closed surface, or a spherical space form. In particular, if M ∞ is noncompact, then every smoothly embedded 2-sphere in M ∞ is separating.

Step 2. The riemannian manifold (M ∞ , g ∞ (0)) has bounded curvature.

Proof. Of course we may assume that M ∞ is noncompact. By passing to the limit, we see that every point p 0 ∈ M ∞ of scalar curvature at least 3 is centre of a (2ε 0 , 2C 0 )-cap or a (not necessarily strong) 2ε 0 -neck. In the sequel, we refer to this fact by saying that the limiting partial flow g ∞ (•) satisfies the weak canonical neighbourhood property.

Let (p k ) be a sequence of points of

M ∞ such that R ∞ (p k , 0) -→ +∞; in particular, d ∞ (p k , x ∞ ) -→ +∞ as k → +∞. Consider segments [x ∞ p k ]
which, after passing to a subsequence, converge to a geodesic ray c starting at x ∞ .

On [x ∞ p k ] we pick a point q k such that R ∞ (q k ) = R∞(p k ) 3C 0 . For sufficiently large k, the point (q k , 0) is centre of a weak (2ε 0 , 2C 0 )-canonical neighbourhood U k . Now the curvatures on U k belong to [(2C 0 ) -1 R ∞ (q k , 0), 2C 0 R ∞ (q k , 0)]. As a consequence, for large k, x ∞ and p k do not belong to U k . By Corollary 6.6 this neighbourhood is a 2ε 0 -neck.

p 1 p 2 p k c q 2 U k q k (M ∞ , g ∞ (0))
x ∞ Lemma 9.3. For k large enough, c traverses U k .

Proof. Assume it does not. Recall that g ∞ (0) is nonnegatively curved. Consider a geodesic triangle x ∞ q k c(t) for t ≥ 10C 0 . Choose k large enough so that ∡(p k x ∞ c(t)) ≤ π/100. Let S -(resp. S + ) be the component of ∂U k which is closest to (resp. farthest from) x ∞ . By comparison with a Euclidean triangle, we see that lim inf t→∞ ∡(x ∞ q k c(t)) ≥ π. Fix t large enough so that this angle is greater than 98/100π. Then q k c(t) intersects S + . The loop γ obtained by concatenating x ∞ q k , q k c(t) and c(t)x ∞ has odd intersection number with S + . This implies that S + is nonseparating. This contradiction proves Lemma 9.3.

We proceed with the proof of Step 2. Pick k 0 large enough so that U k is traversed by c, and let S 0 be the middle sphere of U k 0 . Let a 0 , a ′ 0 ∈ S 0 be two points maximally distant from each other. Call a k , a ′ k the respective intersections of the segments [a 0 c(t)] and [b 0 c(t)] with the middle sphere S k of U k .

a 0 a ′ 0 a k a ′ k
By comparison with Euclidian triangles, when t is large enough, the distance between a k and a ′ k is greater than 1/2d ∞ (a 0 , a ′ 0 ). Thus we have

diam(S k ) ≥ d ∞ (a k , a ′ k ) ≥ 1/2d ∞ (a 0 , a ′ 0 ) = 1/2 diam(S 0 ) . Now the diameter of S k is close to π √ 2R ∞ (q k , 0) -1
and tends to 0 by hypothesis, which gives a contradiction. This completes the proof of Step 2.

Applying again Lemma 9.2 and the distance-curvature Lemma, there exists τ > 0 such that (M k , ḡk (t), (x k , 0)) converges to some Ricci flow on

M ∞ × [-τ, 0]. Define τ 0 := sup{τ ≥ 0, ∃K(τ ), ∀ρ > 0, ∃k(ρ, τ ) s.t. B(x k , 0, ρ) × [-τ, 0]
is unscathed and has curvature bounded by K(τ ) for k ≥ k(ρ, τ )} .

We already know that τ 0 > 0. The compactness theorem A.1 enables us to construct a flow g ∞ (•) on M ∞ × (-τ 0 , 0], which is a pointed limit of the flows ( Mk , ḡk (•)). Moreover, passing to the limit, we see that the scalar curvature of g ∞ is bounded by K(τ ) on (-τ, 0] for all τ ∈ [0, τ 0 ).

Step 3. There exists Q > 0 such that the curvature of g ∞ (t) is bounded above by Q for all t ∈ (-τ 0 , 0].

Proof. We know that g ∞ (t) is nonnegatively curved and has the abovementioned 'weak canonical neighbourhood property'. We show that the conclusion of the curvature-distance theorem holds on M ∞ , at points of scalar curvature > 1. For this, we let p ∈ M ∞ and t ∈ (-τ 0 , 0] be such that R ∞ (p, t) > 1. Then there exists a sequence (p k , t k ), where pk ∈ M k , converging to (p, t) and such that R(p k , t k ) ≥ 1 for k large enough. As a consequence, they satisfy the hypotheses of the curvature-distance theorem as explained in the proof of Step 1. Passing to the limit, we deduce that for every A > 0, there exists Λ(A) > 0 such that for all q ∈ M ∞ ,

R ∞ (q, t) R ∞ (p, t) ≤ Λ d ∞ (p, q, t)R ∞ (p, t) -1 . (10)
Let us estimate the variation of curvatures and distances on M ∞ ×(-τ 0 , 0]. We recall Hamilton's Harnack inequality for the scalar curvature (cf. [KL08, Appendix F])

∂R ∞ ∂t + R ∞ t + τ 0 ≥ 0 , which implies R ∞ (., t) ≤ K(0) τ 0 t + τ 0 .
Ricci curvature, which is positive, satisfies a similar estimate, which implies

const( τ 0 t + τ 0 )g ∞ ≤ ∂g ∞ ∂t ≤ 0 , thus const τ 0 t + τ 0 ≤ ∂d ∞ ∂t (x, y, t) ≤ 0 .
By integration we obtain

|d ∞ (x, y, t) -d ∞ (x, y, 0)| ≤ C .
Since M ∞ is nonnegatively curved, there exists D > 0 such that for all

y ∈ M ∞ , if d ∞ (x ∞ , y, 0) > D, then there exists z ∈ M ∞ such that d ∞ (y, z, 0) = d ∞ (x ∞ , y, 0) and d ∞ (x ∞ , z, 0) ≥ 1.99d ∞ (x, y, 0) (11)
i.e. the points x ∞ , y, z almost lie on a line. (Note that this is true even if M ∞ is compact, because then it is vacuous!)

D x ∞ y z
By comparison with Euclidean space, if D is large enough we have π -1/100 ≤ ∡(x ∞ yz) ≤ π for any such y and z. Observe that since |d ∞ (., ., t)d ∞ (., ., 0)| < C uniformly in t, we can choose D >> C large enough so that, for all t ∈ (-τ 0 , 0], we have

|d ∞ (y, z, t) -d ∞ (x ∞ , y, t)| < 2C and d ∞ (x ∞ , z, t) ≥ 1.98d ∞ (x ∞ , y, t) , thus ∡ t (x ∞ yz) ≥ π -1/50.
Let us show that the scalar curvature of g ∞ (t) is uniformly bounded above on M ∞ \B ∞ (x ∞ , 0, 2D). We argue by contradiction. Suppose that there exists (y i , t i ) such that d ∞ (x ∞ , y i , 0) > 2D and such that R ∞ (y i , t i ) -→ i→+∞ +∞. Each (y i , t i ) has a weak (3ε 0 , C 0 )-canonical neighbourhood. If this neighbourhood is a (3ε 0 , C 0 )-cap, then we let y ′ i be the centre of its 3ε 0 -neck. Since the diameter for g(t i ) of the cap is small (less than 4C 0 R ∞ (y i , t i ) -1 < C for large i), we still have (for large i):

d ∞ (x ∞ , y ′ i , 0) ≥ d ∞ (x ∞ , y ′ i , t i ) -C ≥ d ∞ (x ∞ , y i , t i ) -2C ≥ d ∞ (x ∞ , y i , 0) -3C > D. Furthermore, the curvature of R ∞ (y ′ i , t i ) ≥ 1 3C 0 R ∞ (y i , t i
) tends to +∞. Up to replacing y i by y ′ i , we may assume that there exists a sequence (y

i , t i ) such that d ∞ (x ∞ , y i , 0) > D, R ∞ (y i , t i ) -→ i→+∞ +∞ and (y i , t i ) is centre of a 3ε 0 -neck U i . For each i ∈ N, pick z i ∈ M ∞ satisfying (11). By the above remark, ∡ t i (x ∞ y i z i ) ≥ π -1
50 . The points x ∞ and z i being outside U i , we deduce that x ∞ y i and y i z i each intersect some component ∂U i . Let S i be the middle sphere of U i . This sphere separates x ∞ from z i in the sense that any curve connecting x ∞ to z i passes through S i . Indeed, otherwise the loop obtained by concatenating [x ∞ y i ], [y i z i ] and [z i x ∞ ] would have odd intersection number with S i . As before this leads to a contradiction. Now diam(S i , t i ) -→ 0 since R ∞ (y i , t i ) -→ +∞ as i → +∞. Since g ∞ (t) is nonnegatively curved, distances are nonincreasing in t. As a consequence, diam(S i , 0) -→ 0 i→+∞ . At time 0 the curvature bounds and the hypothesis of κ-noncollapsing imply a uniform lower bound on the injectivity radius. Then for large i, the diameter of S i is less than the injectivity radius of g ∞ (0). This implies that S i bounds a 3-ball which contains neither x ∞ nor z i . Hence we can connect x ∞ to z i by an arc which avoids S i .

This implies that the curvature is bounded outside the g ∞ (0)-ball of radius 2D around x ∞ . We deduce a uniform curvature bound on (-τ 0 , 0] in the ball using Equation (10).

Step 4. We have τ 0 = +∞.

Proof. Consider a subsequence (M k ×(-τ 0 , 0], ḡk (t), (x k , 0)) that converges to (M ∞ × (-τ 0 , 0], g ∞ (t), (x ∞ , 0)). Since the limit has scalar curvature bounded above by Q we deduce that (up to replacing Q by Q+1), for all 0 < τ < τ 0 , for all ρ > 0, there exists k ′ (τ, ρ) ∈ N such that for all k ≥ k ′ (τ, ρ), the parabolic neighbourhood P ḡk (x k , 0, ρ, -τ ) is unscathed and has scalar curvature ≤ Q.

Suppose that τ 0 < +∞ and let 0 < σ < (4C 0 (Q + 2)) -1 . Then up to extracting a subsequence, for every K > 0, there exists ρ = ρ(σ, K) such that P k := P ḡk (x k , 0, ρ, -(τ 0 + σ)) is scathed or does not have curvature bounded above by K.

Set K := 2(Q+2) and ρ :

= ρ(σ, K). If k ≥ k ′ (τ 0 -σ, ρ), then P k is scathed. Indeed, if k ≥ k ′ (τ 0 -σ, ρ), we have R ≤ Q on P ḡk (x k , 0, ρ, -τ 0 + σ))
. If P k is unscathed, the curvature-time lemma (6.2) applied between -τ 0 + σ and -τ 0σ (since 2σ ≤ (2C 0 (Q + 2)) -1 ) implies that R ≤ 2(Q + 2) on P k , which excludes the second alternative.

Thus there exists x ′ k ∈ Bk (x k , 0, ρ), and

t k ∈ [-τ 0 -σ, -τ 0 + σ], assumed to be maximal, such that ḡk (t k ) = ḡk + (t k ) at x ′ k . Since B(x k , 0, ρ) × (t k , 0
] is unscathed, the above argument shows that R ≤ 2(Q + 2) on this set, for all sufficiently large k. This implies an upper bound on the Riemann tensor on this set and hence by Lemma 9.2 the parabolic neighbourhood P ḡk (x k , 0, ρ, t k ) is unscathed. This contradicts the definition of t k .

We can now finish the proof of Proposition B: since τ 0 = +∞, the flow (M ∞ , g ∞ (•)) is defined on (-∞, 0], and has bounded, nonnegative curvature ii. Let r, δ be surgery parameters and g(•) be an (r, δ)-surgical solution. Assume that P 0 = P (x 0 , t 0 , ρ 0 , -ρ 2 0 ) is a scathed parabolic neighbourhood such that | Rm | ρ -2 0 on P 0 . Then P 0 is e -12 κ st /2-noncollapsed.

Remark.

• From (i), we deduce that in order to establish noncollapsing at some point (x, t) on all scales ≤ 1, it suffices to do it on the maximal scale ρ ≤ 1 such that | Rm | ≤ ρ -2 on P (x, t, ρ, ρ -2 ). This observation will be useful later.

• If some metric ball B(y, ρ) is contained in a (ε, C 0 )-canonical neighbourhood which is not ε 0 -round and satisfies | Rm | ρ -2 , then B(y, ρ) is C -1 0 -noncollapsed on the scale ρ by (4).

The proof

We turn to the proof of Proposition C. Let M reg be the set of regular points in spacetime. This is an open, arcwise connected 4-manifold. Likewise we let M sing be the set of singular points in spacetime. Let γ : [t 0 , t 1 ] → t M(t) be a map such that γ(t) ∈ M(t) for every t. Let t ∈ [t 0 , t 1 ]. Here we adopt the convention that M + (t) = M(t) if t is regular.

Definition. One says that γ is continuous at t if there is σ > 0 such that 1) t → γ(t) ∈ M( t) on [ tσ, t) and is left continuous at t 2) t → γ(t) ∈ M + ( t) on ] t, t + σ] and has a right limit at t denoted γ + ( t) 3)Assume t < t 1 . If γ( t) ∈ M reg ( t), then γ( t) = γ + ( t) under the identification of M reg ( t) and M( t) ∩ M + ( t); if γ( t) ∈ S ⊂ S, then γ( t) = γ + ( t) under the identification of S and the corresponding component of ∂M ∩ M + ( t).

In particular, if γ( t) ∈ M sing ( t) \ S for t < t 1 , it is not continuous at t. Indeed, γ( t) disappears at time t.

We say that γ is unscathed if γ(t) ∈ M reg (t) for all t ∈ [t 0 , t 1 ). Otherwise γ is scathed.

We adapt the arguments from the smooth case, replacing Perelman's reduced volume V ([Per02, Section 7]) by:

V reg (τ ) := Y (τ ) τ -3/2 e -ℓ(L exp(v),τ ) J(v, τ )dv, where τ = t 0 -t and Y (τ ) := {v ∈ T x 0 M | L exp(v) : [0, τ ] → M is minimal and unscathed } .
Let (x 0 , t 0 ) be a point. By Lemma 10.1 and the remark following this lemma, we restrict attention to the scale ρ 0 ≤ 1 which is maximal such that | Rm | ≤ ρ -2 0 on P 0 := P (x 0 , t 0 , ρ 0 , -ρ 2 0 ) and assume that P 0 is unscathed. As before we set B 0 := B(x 0 , t 0 , ρ 0 ).

10.3

The case ρ 0 r 100 Lemma 10.2. Let r, ∆, Λ be positive numbers. Then there exists δ = δ(r, ∆, Λ) > 0 with the following property. Let (M(•), g(•)) be an (r, δ)-surgical solution on an interval I = [a, a + ∆] with δ δ and r r on I. Let (x 0 , t 0 ) ∈ M × I and ρ 0 r be such that P 0 := P (x 0 , t 0 , ρ 0 , -ρ 2 0 ) ⊂ M × I is unscathed and | Rm | ≤ ρ -2 0 on P 0 . Let γ be a continuous spacetime curve defined on [t 1 , t 0 ] with t 1 ∈ [0, t 0 ] and such that γ(t 0 ) = x 0 and γ is scathed. Then L t 0 -t 1 (γ) ≥ Λ.

Here L t 0 -t 1 denotes the L-length based at (x 0 , t 0 ), that is

L t 0 -t 1 (γ) = t 0 t 1 √ t 0 -t R(γ(t), t) + | γ(t)| 2 g(t) dt.
Proof. To prove the lemma, it suffices to obtain one of the two inequalities:

t 0 t 1 √ t 0 -t R(γ(t), t) dt ≥ Λ , (12)
t 0 t 1 √ t 0 -t 1 | γ(t)| 2 g(t) dt ≥ Λ + 4∆ 3/2 =: Λ ′ . ( 13
) Indeed L t 0 -t 1 (γ) t 0 t 1 √ t 0 -t R(γ(t), t) dt, hence (12)
implies the lemma.

For (13), this comes from the fact that R ≥ -6, hence

t 0 t 1 √ t 0 -t R dt ≥ -4[τ 3/2 ] t 0 -t 1 0 ≥ -4∆ 3/2 .
Intuitively, those two conditions mean that a curve has large L-length if it has large energy (which is the case if it moves very fast or goes a very long way), or if it stays long enough in an area of large scalar curvature.

Since γ is scathed, it cannot remain in P 0 . We shall make a first dichotomy according to whether γ goes out very fast or not.

Set

α := min r 4Λ ′ 2 , 10 -2 ∈ (0, 10 -2 ) .
Case 1: There exists t ′ ∈ [t 0αρ 2 0 , t 0 ] such that γ(t ′ ) ∈ B 0 .

x 0 t 0 t 0 -αρ 2 0 t 0 -ρ 2 0 γ ρ 0 P 0 γ(t ′ )
Choose t ′ maximal with this property. We then have

t 0 -t ′ 0 | γ| dτ ≤ √ τ | γ| 2 1/2 1 √ τ 1/2 , so τ 1 0 √ τ | γ| 2 dτ ≥ t 0 -t ′ 0 √ τ | γ| 2 dτ ≥ t 0 -t ′ 0 | γ| dτ 2 t 0 -t ′ 0 1 √ τ dτ -1
.

On (t ′ , t 0 ], we have γ ⊂ P 0 . Since P 0 is unscathed, we have

g(t 0 )e -4ρ -2 0 (t 0 -t ′ ) ≤ g(t) ≤ g(t 0 )e 4ρ -2 0 (t 0 -t ′ ) , hence 1 2 g(t 0 ) ≤ e -4α g(t 0 ) ≤ g(t) ≤ e 4α g(t 0 ) ≤ 2g(t 0 ) . Since γ(t ′ ) ∈ B 0 , t 0 -t ′ 0 | γ| g(t 0 -τ ) ≥ 1 √ 2 t 0 -t ′ 0 | γ| g(t 0 ) ≥ ρ 0 √ 2 , so t 0 -t ′ 0 | γ| 2 √ τ dτ ≥ ρ 2 0 2 [2 √ τ ] t 0 -t ′ 0 -1 ≥ ρ 0 4 √ α ≥ r 4 √ α .
By choice of α, this last quantity is bounded below by Λ ′ . This shows that γ satisfies Inequality (13).

Remark. In this case, there is no constraint on δ.

Case 2: For all t ∈ [t 0αρ 2 0 , t 0 ], γ(t) ∈ B 0 .

Closeness with the standard solution implies

t+θh 2 t √ t 0 -tR dt const st 2 t+θh 2 t √ t 0 -t h -2 1 -(t -t)h -2 dt const st αρ 2 0 2 t+θh 2 t h -2 1 -(t -t)h -2 dt = const st αρ 2 0 2 θ 0 1 1 -u du -const st √ αr 2 ln(1 -θ) Λ ′ ,
for θ close enough to 1, depending only on r, ∆, Λ. We deduce

t 0 t 1 √ t 0 -tR dt = t+θh 2 t √ t 0 -tR dt + t t 1 √ t 0 -tR dt + t 0 t+θh 2 √ t 0 -tR dt Λ ′ -4∆ 3/2 = Λ.
Hence Inequality (12) holds. Fix θ := θ(r, ∆, Λ) such that this condition is satisfied.

Subcase ii: There exists t ′ ∈ [ t, t + θh 2 ] such that γ(t ′ ) ∈ B(x, t, Ah). We assume that t ′ is minimal with this property. By the same argument as before, P ′ := P + (x, t, Ah, t ′ ) is unscathed, and A -1 -close to the standard solution by the persistence theorem. As before, this implies that for all s, s ′ ∈ ( t, t ′ ], g(s) ≤ e C 1-θ g(s ′ ) ,

where C is a universal constant. Thus we have

t ′ t √ τ | γ| 2 g(t) ≥ e -C 1-θ | γ| g( t) 2 1 √ τ -1
.

Since t ′ ≤ t 0αρ 2 0 we bound τ from below by αρ 2 0 on [ t, t ′ ], so

t ′ t √ τ | γ| 2 ≥ e -C 1-θ (Ah) 2 αρ 2 0 (t ′ -t) -1 ≥ e -C 1-θ (Ah) 2 r 10 √ α 1 θh 2 = e -C 1-θ √ α rA 2 10θ .
Fixing A large enough, Inequality (13) holds.

A consequence of the previous lemma is the following result (see [KL08, Lemmas 78.3 and 78.6] or [BBB + ] for more details).

Lemma 10.3. Let r, ∆, Λ be positive numbers. There exists δ := δ(r, ∆, Λ) with the following property. Let g(•) be a (r, δ)-surgical solution defined on [a, a + ∆] such that r r and δ ≤ δ. Let (x 0 , t 0 ) and ρ 0 r be such that P 0 := P (x 0 , t 0 , ρ 0 , -ρ 2 0 ) is unscathed and | Rm | ≤ ρ -2 0 on P 0 . Then i. ∀(q, t) ∈ M × [a, a + ∆], if ℓ(q, t 0t) < Λ, then there is an unscathed minimising L-geodesic γ connecting x 0 to q.

ii. ∀τ > 0, min q ℓ(q, τ ) ≤ 3/2 and is attained.

We come back to the proof of Proposition C. Recall that P 0 is unscathed, and satisfies | Rm | ≤ ρ -2 0 . The arguments of [Per02, §7] apply to unscathed minimising L-geodesics. In particular, if γ(τ ) = L τ exp (x 0 ,t 0 ) (v) is minimising and unscathed on [0, τ 0 ], then τ -3/2 e -ℓ(v,τ ) J(v, τ ) is nonincreasing on [0, Lower bound for V reg (t 0 ) Monotonicity of V reg (τ 0 ) implies

V reg (τ 0 ) ≥ V reg (t 0 ) .
We are going to bound V reg (t 0 ) from below as a function of volB(q 0 , 0, 1), where q 0 ∈ M. Set Λ := 21 and apply Lemma 10.3 with parameter δ(r, T, Λ). There exists q 0 ∈ M such that ℓ(q 0 , t 0 -1 16 ) ≤ 3 2 and the first part of Lemma 10.3 gives us a minimising curve γ connecting x 0 to q 0 realising the minimum ℓ(q 0 , t 0 -1/16). Let q ∈ B(q 0 , 0, 1). Consider a curve γ obtained by concatenating some g(0)-geodesic from (q, 0) to (q 0 , 1/16) with some minimising L-geodesic between (q 0 , 1/16) and (x 0 , t 0 ). We have ℓ(q, t 0 ) = L(q, t 0 ) 2 √ t 0 ≤ 1 2 √ t 0 L(q 0 , t 0 -1/16) + 1 2 √ t 0 t 0 t 0 -1/16 √ τ (R g(t 0 -τ ) + | γ(τ )| 2 g(t 0 -τ ) )dτ , (14) which leads to ℓ(q, t 0 ) ≤ 3 2 + 1 2 t 0 t 0 -1/16

(12 + e 1/2 | γ(τ )| 2 g(0) )dτ since for s ∈ [0, 1/16] (i.e. τ ∈ [t 0 -1/16, t 0 ]) the metrics g(s) satisfy Rm ≤ 2, hence R ≤ 12, and are 1/2-Lipschitz equivalent. We obtain ℓ(q, t 0 ) ≤ 3 2 + 1 2 ( 1 16 12 + 16e 1/2 d 2 g(0) (q, q 0 )) ≤ 20 .

Hence ℓ(q, t 0 ) < Λ, and Lemma 10.3 i) re-gives an unscathed minimising L-geodesic γ connecting x 0 to q. Hence q = γ(t 0 ) = L t 0 exp(ṽ) for ṽ ∈ Y (t 0 ). This shows that L t 0 exp(Y (t 0 )) ⊃ B(q, 0, 1). Moreover, we have {ṽ ∈ Y (t 0 ) ; L t 0 exp(ṽ) ∈ B(q, 0, 1)}, ℓ ≤ 10, which implies V reg (t 0 ) = Y (t 0 ) t -3/2 0 e -ℓ(v,t 0 ) J(v, t 0 )dv ≥ B(q 0 ,0,1) t -3/2 0 e -20 dv g(0)

≥ T -3/2 e -20 vol g(0) B(q 0 , 0, 1) . Theorem 11.4. There exist sequences r k , δ k , κ k > 0 such that for any complete normalised riemannian 3-manifold (M 0 , g 0 ), there exists a surgical solution (M(•), g(•)) defined on [0, +∞), satisfying the initial condition (M(0), g(0)) = (M 0 , g 0 ), and such that for every nonnegative integer k, the restriction of (M(•), g(•)) to [k, k + 1] is an (r k , δ k , κ k )-surgical solution.

Moreover, if (M 0 , g 0 ) is endowed with a properly discontinuous isometric action of some group Γ, then the surgical solution can be made Γ-equivariant. In addition, if the action of Γ on M 0 is free, then the action on each M(t) can be chosen to be free.

This follows from iteration of Theorem 5.3. Indeed, assuming the parameters r k , δ k , κ k > 0 are known, we deduce from Θ k := Θ(r k , δ k ) a bound for the sectional curvature Q k . From this and the κ k -noncollapsing property, we deduce a lower bound for volumes of balls of radius at most Q -1/2 k . This gives a lower bound ρ k for the injectivity radius of every metric, in particular the metric g(k + 1).

The addendum about equivariance follows as explained in the proof of Theorem 11.2.

Open questions

The first question asks whether the hypotheses of Corollary 11.3 are necessary.

Question. Let M be a connected, orientable 3-manifold which admits a complete riemannian metric of uniformly positive scalar curvature. Is M a connected sum of spherical manifolds and copies of S 2 × S 1 ?

Next we consider what happens when we relax the hypothesis on the scalar curvature from uniform positivity to positivity. This class is significantly wider, e.g. it includes S 1 × R 2 . One could even relax the condition further to nonnegativity.

Question (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.

A Hamilton's compactness theorem

A pointed evolving metric is a triple (M, {g(t)} t∈I , (x 0 , t 0 )) where M is a manifold, g(•) is an evolving metric on M, and (x 0 , t 0 ) belongs to M × I. We say that a sequence of pointed evolving metrics (M k , {g k (t)} t∈I , (x k , t 0 )) converges smoothly to a pointed evolving metric (M ∞ , {g ∞ (t)} t∈I , (x ∞ , t 0 )) if there exists an exhaustion of M by open sets U k , such that x ∈ U k for all k, and smooth embeddings ψ k : U k → M k sending x to x k and such that ψ * k g k (•)g(•) and all its derivatives converge to zero uniformly on compact subsets of M × I. Remark. If g(t) is defined on [a, b], one can take t 0 = a if one has also uniform bounds on the derivatives of the curvature operator at time t 0 , i.e. if for any ρ > 0, for any integer p, sup k∈N sup B(x k ,t 0 ,ρ)×{t 0 } |∇ p Rm | < +∞.

B Partial Ricci flows

Definition. Let (a, b] a time interval. A partial Ricci flow4 on U × (a, b] is a pair (P, g(•, •)), where P ⊂ U × (a, b] is an open subset which contains U × {b} and (x, t) → g(x, t) is a smooth map defined on P such that the restriction of g to any subset V × I ⊂ P is a Ricci flow on V × I.

Inspection of the proof of Theorem A.1 shows that the following natural extension holds.

Theorem B.1 (Local compactness for flows). Let (U k , {g k (t)} t∈(a,0] , (x k , 0)) be a sequence of pointed Ricci flows of the same dimension. Suppose that for some ρ 0 ∈ (0, +∞], all the balls B(x k , 0, ρ) of radius ρ < ρ 0 are relatively compact in U k and that the following holds : i. For any ρ ∈ (0, ρ 0 ), there exists Λ(ρ) < +∞ and τ (ρ) > 0 such that | Rm | < Λ(ρ) on all P (x k , 0, ρ, -τ (ρ)).

ii.

inf k∈N inj(U k , g k (0), x k ) > 0.
Then there is a riemannian ball B(x ∞ , ρ 0 ) of the same dimension such that the pointed sequence (B(x k , 0, ρ 0 ), g k (•), x k ) subconverges smoothly to a partial Ricci flow g ∞ (•) defined on ρ<ρ 0 (B(x ∞ , ρ) × (-τ (ρ), 0]). Moreover, if ρ 0 = +∞ then for any t ∈ [sup ρ -τ (ρ), 0], g ∞ (t) is complete.

  then g(•) has curvature pinched toward positive at time 0. Next we state a result due to Hamilton and Ivey in the compact case. For a proof of the general case, see [CCG + 08, Section 5.1]. Proposition 3.1 (Hamilton-Ivey pinching estimate). Let a, b be two real numbers such that 0 ≤ a < b. Let (M, {g(t)} t∈[a,b] ) be a complete Ricci flow with bounded curvature. If g(•) has curvature pinched toward positive at time a, then {g(t)} t∈[a,b] has curvature pinched toward positive.

  g(•)) to a surgical solution defined on [T A , b max ], with R max (b max ) ≤ Θ. Condition (NC) κ is still satisfied on this closed interval by Lemma 5.5. Hence we can apply Proposition B, which implies that Property (CN) r is satisfied on [T A , b max ]. We thus obtain an (r, δ, κ)-surgical solution on the closed interval [T A , b max ].

Notation

  If (M(•), g(•)) is a piecewise C 1 evolving manifold defined on some interval I ⊂ R and [a, b] ⊂ I, we call restriction of g to [a, b] the evolving manifold

  Since B ḡ(x, 0, ρ) × (s, 0] is unscathed with | Rm | ≤ K, by the distancedistorsion Lemma 9.1 we have e -4Kτ ≤ ḡ(0)/ḡ(s) ≤ e 4Kτ , which implies d ḡ(s) (x, z) ≤ e 2Kτ ρ. Since y ∈ U, R ḡ(y, s) ≤ 2R ḡ(z, s) ≤ 12K .

0 , τ 0 = κ 1/3 ρ 2 0 .

 00 τ 0 ]. Define Y (τ ) := {v ∈ T x 0 M ; L exp(v) : [0, τ ] -→ M is minimising and unscathed } . It is easy to check that τ ≤ τ ′ ⇒ Y (τ ) ⊃ Y (τ ′ ). Then we set V reg (τ ) := Y (τ ) τ -3/2 e -ℓ(v,τ ) J(v, τ )dv .This function is nondecreasing on [0, t 0 ]. We shall adapt the proof of κnoncollapsing in the smooth case, replacing V by V reg . Setκ := vol g(t 0 ) B 0 ρ 3Upper bound on V reg (τ 0 ): As in the smooth case, we getL τ exp({v ∈ Y (τ ) ; |v| ≤ 1 10 κ -1/6 }) ⊂ B 0 thus I := {v∈Y (τ ) ; |v|≤ 1 10 κ -1/6 } τ -3/2 e -ℓ(v,τ ) J(v, τ )dv ≤ e Cκ 1/6 √ κ and I ′ := {v∈Y (τ ) ; |v|≥ 1 10 κ -1/6 } τ -3/2 e -ℓ(v,τ ) J(v, τ )dv ≤ {v∈Y (τ ) ; |v|≥ 1 10 κ -1/6 } lim τ →0 (τ -3/2 e -ℓ(v,τ ) J(v, τ ))dv ≤ e -1 10 κ -1/6 .In conclusion,I + I ′ ≤ e Cκ 1/3 √ κ + e -1 10 κ -1/6 ≤ C √ κfor κ ≤ κ(3) and some universal C.

  Theorem A.1 (Hamilton's compactness). Let (M k , {g k (t)} t∈(a,b] , (x k , t 0 )) be a sequence of complete pointed Ricci flows of the same dimension. Assume: i. For all ρ > 0 , supk∈N sup B(x k ,t 0 ,ρ)×(a,b] | Rm | < +∞ ii. inf k∈N inj(M k , g k (t 0 ), x k ) > 0.Then (M k , {g k (t)} t∈(a,b] , (x k , t 0 )) converges smoothly to a complete Ricci flow of the same dimension, defined on (a, b].

  Assume that t is closest to t 0 with this property. If t > t 0 then x ∈ M sing (t) and disappears at time t unless x ∈ ∂M sing (t) or if the component of M sing (t) which contains x is a sphere S ∈ S. If t < t 0 , then x ∈ M + (t) \ M reg (t) is in one of the 3-balls that are added at time t.

	b] is unscathed, then t → g(t) evolves smoothly by the Ricci flow equation on all of X × [a, b].
	2) Assume that X × [a, b] is scathed. Then there is t ∈ [a, b) and x ∈ X such that x / ∈ M reg (t).

  (Local curvature-time lemma). Let (U, g(•)) be a Ricci flow defined on [t 1 , t 2 ]. Let Q 1, C > 0, x ∈ U and set Q x =| R(x, t 2 ) | +Q.Suppose that there exists t ∈ [t 1 , t 2 ] such that R(x, t) 2Q x , and that

					xy]
	where
	( * ) holds at each point of scalar curvature at least Q. Then d(x, y) | ∇R | CR 3/2	2C	1 √	Qx .
	Lemma 6.2 |	dR dt	| CR 2	( * * )

See below for the precise definition of capping-off.

However, there have been attempts to generalise the theory beyond this framework, see e.g.[START_REF] Xu | Short-time existence of the ricci flow on noncompact riemannian manifolds[END_REF][START_REF] Simon | Ricci flow of non-collapsed 3-manifolds whose ricci curvature is bounded from below[END_REF].

We use 'strong neck' to denote a subset of M (t 0 ), rather than a subset of spacetime as other authors do.

This definition differs from that of local Ricci flow introduced by D. Yang[START_REF] Yang | Convergence of Riemannian manifolds with integral bounds on curvature[END_REF].

Proof of Proposition A

Recall the statement:

Proposition A. There exists a universal constant δA > 0 having the following property: let r, δ be surgery parameters, a, b be positive numbers with a < b, and {(M(t), g(t))} t∈(a,b] be an (r, δ)-surgical solution. Suppose that δ ≤ δA , and R max (b) = Θ.

Then there exists a riemannian manifold (M + , g + ), which is obtained from (M(•), g(•)) by (r, δ)-surgery at time b, and in addition satisfies: i. g + has φ b -almost nonnegative curvature;

ii. R min (g + ) ≥ R min (g(b)).

Throughout this section we shall work in the riemannian manifold (M(b), g(b)).

In particular all curvatures and distances are taken with respect to this metric.

Let G (resp. O, resp. R) be the set of points of M(b) of scalar curvature less than 2r -2 , (resp. ∈ [2r -2 , Θ/2), resp. ≥ Θ/2.)

For brevity, we call cutoff neck a strong δ-neck centred at some point of scalar curvature h -2 . Note that cutoff necks are contained in O, and have diameter and volume bounded by functions of h, δ-alone.

Lemma 7.6. There exists a locally finite collection {N i } of pairwise disjoint cutoff necks such that any connected component of M(b) \ i N i is contained in either G ∪ O or R ∪ O.

Proof. By Zorn's Lemma, there exists a maximal collection {N i } of pairwise disjoint cutoff necks. Such a collection is automatically locally finite, e.g. because if K is any compact subset, all cutoff necks that meet K are contained in the h(2δ -1 + 1)-neighbourhood of K, which has finite volume.

Suppose that some component X of M(b) \ i N i contains at least one point x ∈ G and one point z ∈ R. Since X is a closed subset of M(b), there exists a geodesic path γ in X connecting x to z.

Claim. The intersection of γ and ∂X is empty.

Proof of the claim. First observe that if y is a point of ∂X, then y has a canonical neighbourhood U. This neighbourhood cannot be a cap, because then U would contain the whole of X, which would imply that X ⊂ O. Hence U is a neck.

If such a point y belonged to γ, then by Lemma 6.5 the neck U would be traversed by γ. This contradicts the fact that γ ⊂ X.

Suppose that R ḡ(y, s) 1/2τ . Seeking a contradiction, assume that s 1 := s + θR ḡ (y, s) -1 ≤ 0 and apply the persistence theorem up to this time. By Proposition 4.3, R min (t) const st (1t) -1 , hence we have

On the other hand, R ḡ (x, s 1 ) 6K, which gives a contradiction for sufficiently large k. This proves the claim.

Denote by g(•) the parabolic rescaling of ḡ by R ḡ(y, s) at time s:

By the conclusion of the persistence theorem, there exists a diffeomorphism ψ :

By the above claim, the minimum is |s|R ḡ(y, s) := s ′ . Set x ′ := ψ -1 (x). Proposition 4.3 implies that for every ǫ > 0, there exists C st (ε) such that any point (x ′ , t) of the standard solution has an (ε, C st (ε))-canonical neighbourhood unless t < 3/4 and x ′ / ∈ B(p 0 , 0, ε -1 ). Let us choose ε := ε 0 β/2 << ε and C st := C st (ε). There are again two possibilities.

Case 1 The point (x ′ , s ′ ) has an (ε, C st )-canonical neighbourhood U ′ ⊂ B(x ′ , s ′ , 2C st R(x ′ , s ′ ) -1/2 ), where R(x ′ , s ′ ) is the scalar curvature of the standard solution at (x ′ , s ′ ). The A -1 -closeness between g 0 (s ′ ) and ψ * g(s ′ ) gives

On the other hand,

Therefore ψ(U ′ ) is a (2ε, 2C st )-canonical neighbourhood for (x, 0), hence an (ε 0 , C 0 )-canonical neighbourhood for this point.

operator. Moreover, the rescaled evolving metric ḡk (•) is κ-noncollapsed on scales less than R(x k , b k )r 0 , so passing to the limit we see that g ∞ (•) is κ-noncollapsed on all scales. The metric g ∞ (•) is not flat since it has scalar curvature 1 at the point (x ∞ , 0). This shows that (M ∞ , g ∞ (•)) is a κ-solution. By Theorem 4.1 and the choice of constants in Subsection 4.3, every point of M ∞ has an ( ε 0 2 , C 0 2 )canonical neighbourhood. Hence for sufficiently large k, (x k , 0) has an (ε 0 , C 0 )canonical neighbourhood. This contractiction finishes the proof of Proposition B.

Proof of Proposition C

We recall the statement:

Q 0 , has injectivity radius at least ρ 0 , φ A -almost nonnegative curvature and satisfies R min (g 0 ) ≥ -6/(4T A + 1). Then g(•) satisfies (NC) κ .

Note that by Cheeger's theorem and standard estimates on Ricci flow, there exists a constant κ norm depending only on the normalisation of the initial condition, i.e. Q 0 , ρ 0 , such that (M(•),

Preliminaries

Let v k (ρ) denote the volume of a ball of radius ρ in the model space of constant sectional curvature k and dimension n.

Let κ > 0. One says that a Riemannian ball

We recall the following elementary lemma from [BBB + , Section 8.1].

v -1 (1) . The same property holds for P (x, t, ρ ′ , -ρ ′2 ) ⊂ P (x, t, ρ, -ρ 2 ). Since γ is scathed, there exists (x, t) such that γ( t) / ∈ M reg ( t). Since P 0 is unscathed, we have t < t 0αρ 2 0 . Since γ is continuous and defined after t, we have γ( t) ∈ ∂M sing (t) ⊂ S(t). Assume that t is maximal for this property. We have R(x, t) ≈ h -2 , where, for the sake of simplicity we set h := h( t). We may choose δ small enough (depending on r) so as to make h so small that R(x, t) is strictly greater than 12r -2 12ρ -2 0 . For constants θ ∈ [0, 1) and A >> 1 to be chosen later, we set

and take δ δper (A, θ, r), so that Theorem 8.1 applies. In particular, we have R ≥ 1 2 h -2 > 6ρ -2 0 on P . This implies that P 0 ∩ P = ∅. We distinguish two subcases.

Then by Theorem 8.1, P is unscathed. Indeed, otherwise

10.4 The case ρ 0 r 100

In this case ρ 0 is below the scale of the canonical neighbourhood for all t in the interval and hence any point with scalar curvature greater than ρ -2 0 has a canonical neighbourhood. At such a point the noncollapsing is given by this neighbourhood if it is not ε-round.

Since ρ 0 < 1, there exists (y, t) ∈ P0 such that | Rm(y, t)| = ρ -2 0 . Hence we have | Rm(y, t)| r -2 10 6 .

Since {g(t)} has curvature pinched toward positive,

Hence (y, t) has an (ε 0 , C 0 )-canonical neighbourhood U.

Case 1: U is not ε 0 -round Let us show that B(x 0 , t, e -2 ρ 0 ) ⊂ U. This is clear if U is closed, so we only have to deal with the cases of necks and caps. By the curvature bounds on P0 we have d t (x 0 , y) e 2 ρ 0 and B(x

If U is an (ε 0 , C 0 )-cap, write it U = V ∩ W where V is a core. Let γ : [0, 1] → B0 be a minimising g(t 0 )-geodesic connecting y to x 0 . If x 0 / ∈ V , let s ∈ [0, 1] be maximal such that γ(s) ∈ ∂V . Since γ(s) ∈ B 0 , we have R(γ(s), t) 6ρ -2 0 and we deduce that d(γ(s), ∂U ) ≥ ( √ 6ε 0 ) -1 ρ 0 . As d t (γ(s), x 0 ) e 2 ρ 0 we get B(x 0 , t, e -2 ρ 0 ) ⊂ U.

Comparing this to Equation (4), we see that vol g(t) B(x 0 , t, e -2 ρ 0 ) C -1 0 (e -2 ρ 0 ) 3 . By estimates on distortion of distances and volume as in the proof of Lemma 10.1, we conclude that vol g(t 0 ) B 0 C -1 0 e -18 ρ 3 0 .

Case 2: U is ε 0 -round Note that the method of Case 1 applies equally well if U is homeomorphic to S 3 or RP 3 , so we assume it is not the case.

The only thing we have to do is to prove that there are only finitely many possible topologies for U. For simplicity of notation we assume (x 0 , t 0 ) = (y, t), i.e. the point (x 0 , t 0 ) has an ε 0 -round canonical neighbourhood U, and | Rm(x 0 , t 0 )| ≥ 1000r -2 . Lemma 10.4. There exists t ′ 0 < t 0 such that

) is ε 0 -round;

• letting ρ ′ 0 be defined at (x 0 , t ′ 0 ) in the obvious way, we have 2r ≥ ρ ′ 0 ≥ r 2 . Proof. Let t ′′ 0 < t 0 be minimal such that U is unscathed and for every t ∈ [t ′′ 0 , t 0 ], (U, g(t)) is ε 0 -round and R ≥ r -2 on (U, g(t)). We claim that R min = r -2 on (U, g(t ′′ 0 )). Indeed by continuity R ≥ r -2 on (U, g(t ′′ 0 )). Hence (x 0 , t ′′ 0 ) has a canonical neighbourhood V . By continuity, (U, g(t ′′ 0 )) is 2ε 0 -round, so V = U; since we have excluded S 3 and RP 3 , we deduce that V is in fact ε 0round. Since ε 0 -roundness is an open property, it follows that if R min > r-2 on (U, g(t ′′ 0 )) then t" 0 is not minimal. This proves the claim.

) (comparing with the evolving round metric one can find t ′ 0 close to t" 0 + 5 4 r 2 ). It follows that the maximal radius

2 ) =: P 0 with P 0 unscathed, is close to r.

Since ρ ′ 0 ≥ r/2 we can argue as in subsection 10.3 to get uniform noncollapsing at (x 0 , t ′ 0 ) on the unit scale. As (U, g(t ′ 0 )) is ε 0 -homothetic to (U, g(t 0 )) and ρ ′ 0 ≤ 2r < 1, we also have uniform noncollapsing at (x 0 , t 0 ) on the unit scale.

11 Generalisations and open questions

Consequences and generalisations

First we state a finiteness result which follows immediately from Theorem 5.2 and Corollary 2.2.

Corollary 11.1. Let R 0 , Q, ρ be positive numbers. Then the class of prime 3manifolds admitting complete riemannian metrics of scalar curvature greater than R 0 , sectional curvature bounded in absolute value by Q, and injectivity radius greater than ρ is finite up to diffeomorphism.

Remark that the primeness hypothesis is necessary: otherwise, one could have, say, a connected sum of arbitrarily many copies of the same manifold. The key point is that the geometric bounds considered here do not imply any diameter bound (nor compactness of the manifold for that matter.) Hence Corollary 11.1 is not a purely geometric finiteness theorem, but rather a mixed geometrico-topological finiteness theorem.

Next we discuss an equivariant version of our main technical theorem.

Definition. Let (M(•), g(•)) be a surgical solution defined on some interval I. Let Γ be a group endowed with an action on each M(t) for t ∈ I, which is constant in between singular times. We say that (M(•), g(•)) is Γ-equivariant if for each t, the action of Γ on M(t) is isometric, and for each singular time t, the union of all 2-spheres along which surgery is performed is Γ-invariant.

Theorem 11.2. Let M be an orientable 3-manifold. Let g 0 be a complete riemannian metric on M which has bounded geometry. Let Γ be a group acting properly discontinuously on M by isometries for g 0 . Then there exists a complete surgical solution (M(•), g(•)) of bounded geometry defined on [0, +∞), with initial condition (M(0), g(0)) = (M, g 0 ), and such that there is for each t a properly discontinuous action of Γ on M(t) such that (M(•), g(•)) is Γ-equivariant, and such that if t is a singular time and x a point belonging to some disappearing component, then (x, t) has an (ε 0 , C 0 )-canonical neighbourhood. Furthermore, if the action of Γ on M is free, then one can ensure that for each t, the action of Γ on M(t) is also free.

Proof. We repeat the proof of Theorem 5.3, paying attention to equivariance with respect to the group Γ. By the Chen-Zhu uniqueness theorem [START_REF] Chen | Uniqueness of the Ricci flow on complete noncompact manifolds[END_REF], Ricci flow automatically preserves the symmetries of the original metric, so the only thing to check is that surgery can be done equivariantly. For this we can apply [DL09, Lemma 3.9]. Note that the constant ǫ appearing in that paper is a priori smaller than our ε 0 . However, it is easy to check that if we replace ε 0 by some smaller positive number ε ′ 0 in the proof of Theorem 5.3 and subsequently the constants β 0 and C 0 by the appropriate constants β ′ 0 and C ′ 0 , then the proof goes through without changes. For the addendum where it is assumed that the action of Γ is free, there is an additional point to check: that surgery can be done so that the action of Γ on the post-surgery manifold is still free. For simplicity, we are going to explain this in a riemannian setting, ignoring the issue of strong necks, which is irrelevant here.

Let (X, g) be a 3-manifold with an isometric, free, properly discontinuous action of Γ and {N i } be a Γ-invariant, locally finite collection of pairwise disjoint δ-necks in X. Let (Y, g) be the quotient riemannian manifold X/Γ.

Suppose first that for each N i and each nontrivial element γ ∈ Γ we have N i ∩ γN i = ∅. Then the collection {N i } projects to a locally finite collection of pairwise disjoint δ-necks in Y . Hence we can do metric surgery on Y , obtaining a riemannian manifold (Y + , g + ). We then lift the construction, getting a riemannian manifold (X + , g+ ) which on the one hand is obtained from (X, g) by metric surgery on {N i }, and on the other hand inherits a free, properly discontinuous, isometric action of Γ.

Thus we are done unless there exists i and γ such that N i ∩ γN i = ∅. In this case, N i is invariant by γ. Since γ acts freely, it must act on N i by an involution, so that N i projects to a cap C ⊂ Y diffeomorphic to a punctured RP 3 . In this case, C contains, say a 4δ-neck whose preimage in X contains two 4δ-necks interchanged by γ. Thus, up to replacing δ by 4δ, we can apply the construction of the first paragraph.

Corollary 11.3. Let M be a connected, orientable 3-manifold which carries a complete metric g of uniformly positive scalar curvature. Assume that the riemannian universal cover of (M, g) has bounded geometry. Then M is a connected sum of spherical manifolds and copies of S 2 × S 1 .

Proof. We apply Theorem 11.2 to the universal cover of (M, g) endowed with the action of Γ := π 1 (M). Let ( M (•), g(•)) be a surgical solution satisfying the conclusion of that theorem. By Corollary 2.2, this surgical solution must be extinct. Thus M is a connected sum of metric quotients of the disappearing components of ( M(•), g(•)). There remains to check that such quotients are themselves connected sums of spherical manifolds and copies of S 2 × S 1 .

We use the fact that the disappearing components are covered by canonical neighbourhoods, and the action of Γ on them is isometric. Let X be such a component. Remark that X is simply-connected, since the van Kampen theorem implies that surgery along 2-spheres on a simply-connected 3-manifold produces simply-connected 3-manifolds. If X is compact, then by Perelman's Geometrisation Theorem, X is diffeomorphic to the 3-sphere, and its quotients are spherical manifolds.

If X is noncompact, then it is diffeomorphic to S 2 ×R or R 3 . In the former case, it is an exercise in topology (cf. [START_REF] Scott | The geometries of 3-manifolds[END_REF]) to show that the quotient can only be S 2 × R itself, a punctured RP 3 , S 2 × S 1 , or a connected sum of two copies of RP 3 .

In the latter case, we obviously need to use the geometry. As in [DL09, Section 3], we consider the open subset T consisting of all points that are centres of ε 0 -necks. Since X is diffeomorphic to R 3 , T is an ε 0 -tube, and its complement C is the core of an ε 0 -cap and diffeomorphic to the 3-ball. By definition, T is automatically invariant by any isometry. Hence C is also invariant by any isometry. Thus by the Brouwer fixed point theorem, X does not admit any nontrivial free isometric group action.

Here is a more precise theorem which may be useful for subsequent applications: