R Ésum

Athanase Papadopoulos

Stéphane Sabourau

La Je Remercie Également

Anais Secrétaire

Delgado

Alaa Cosmin

David

Houssam Jean-Maxime Hien

Khaled, Laurent Juan Jérémy

les responsables du projet

Keywords: Systole, aire, métrique extrémale, métrique de Finsler, bouteille de Klein, ruban de Möbius, surface de genre deux Systole, area, extremal metric, Finsler metric, Klein bo le, Möbius band, genus two surface. vi

En 1949, C. Loewner a demontré dans un travail non publié que si T 2 est un tore Riemannien, alors aire(T 2 ) ≥ √ 3 2 sys(T 2 ) 2 où sys(T 2 ) est la systole du tore T 2 , i.e. la longeur du plus court lacet non contractile de T 2 . De plus, l'égalité est a einte si et seulement si le tore est plat hexagonal. Ce résultat a donné naissance à la géométrie systolique. Dans ce e thèse, nous étudions des inégalités de type systolique portant sur les longueurs minimales de di érentes courbes et pas seulement la systole. Dans un premier temps, nous démontrons trois inégalités géométriques optimales conformes sur la bouteille de Klein reliant l'aire au produit des longueurs des plus courts lacets noncontractiles dans des classes d'homotopie libres di érentes. Pour chaque classe conforme, nous décrivons la métrique extrémale réalisant le cas d'égalité. Nous établissons ensuite des inégalités géométriques optimales sur le ruban de Möbius muni d'une métrique de Finsler. Ces inégalités géométriques relient la systole et la hauteur du ruban de Möbius à son volume de Holmes-ompson. Nous en déduisons une inégalité systolique optimale sur la bouteille de Klein munie d'une métrique de Finsler avec des symétries. Nous décrivons également une famille de métriques extrémales dans les deux cas. Dans le troisième travail, nous démontrons une inégalité systolique critique sur la surface de genre deux. Plus précisément, il est connu que la surface de genre deux admet une métrique Riemannienne plate à singularités coniques qui est extrémale parmi les métriques à courbure nonpositive pour l'inégalité systolique. Nous montrons que ce e métrique est en fait critique pour des variations lentes de métriques, ce e fois-ci sans hypothèse de courbure, pour un autre problème systolique portant sur les longueurs des plus courts lacets non contractiles dans certaines classes d'homotopie libres données. Ces classes d'homotopie correspondent aux lacets systoliques et deux-systoliques de la surface extrémale.

INTRODUCTION

We are interested in studying optimal geometric inequalities relating the area of a closed Riemannian surface Σ to the lengths of the shortest closed geodesics in certain homotopy classes.

A typical example is given by the systolic inequalities, which relate the area of the surface to the systole, that is, the smallest length of a noncontractible loop in Σ. e rst known systolic inequality is due to C. Loewner in 1949. He proved that every Riemannian two-torus T 2 satis es

area(T 2 ) ≥ √ 3 2 sys(T 2 ) 2 ,
where sys(T 2 ) denotes the systole of T 2 . Furthermore, the equality holds if and only if the torus is at hexagonal. C. Loewner did not publish his result, however it was mentioned by his student P. Pu, cf. [START_REF] Pu | Some inequalities in certain non-orientable Riemannian manifolds[END_REF], who established a sharp systolic inequality on the projective plane RP 2 . In this case, the equality is a ained precisely by the round metrics. More than thirty years later, C. Bavard, cf. [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF] (see also [START_REF] Sakai | A proof of the isosystolic inequality for the Klein bo le[END_REF]) proved a sharp systolic inequality on the Klein bo le.

Here, the extremal metrics are not smooth. ese are the only manifolds for which an optimal systolic inequality is known. Optimal systolic-like inequalities, i.e., inequalities that relate the area to the product of the lengths of the shortest loops or arcs in di erent (relative) homotopy classes are known only for the two-torus T 2 , cf. [START_REF] Keen | An extremal length on a torus[END_REF], the Klein bo le K, cf. [START_REF] Bavard | Une remarque sur la géométrie systolique de la bouteille de Klein[END_REF][START_REF] El Mir | Conformal geometric inequalities on the Klein bo le[END_REF], and the Mobius band M, cf. [START_REF] Pu | Some inequalities in certain non-orientable Riemannian manifolds[END_REF][START_REF] Bla Er | Zur Riemannschen Geometrie im Grossen auf dem Möbiusband[END_REF][START_REF] Bavard | Une remarque sur la géométrie systolique de la bouteille de Klein[END_REF]. With the exception of some recent proofs of systolic inequalities, cf. [START_REF] Ivanov | On two-dimensional minimal llings[END_REF][START_REF] Sabourau | Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic[END_REF], and systolic-like inequalities, cf. [START_REF] Sabourau | Optimal systolic inequalities on Finsler Mobius bands[END_REF], on the projective plane RP 2 , the two-torus T 2 and the Mobius band M, all the other known proofs of the aforementioned optimal geometric inequalities require the uniformization theorem as a main tool.

is thesis is divided into three chapters. In these chapters, we study various optimal systolic-like inequalities for di erent surfaces: the Klein bo le, the Mobius band, the torus and the genus two surface. Each chapter relies on speci c techniques: extremal length methods in the rst chapter, Finsler geometry in the second chapter and in nitesimal calibrating arguments in in nitedimensional spaces in the third chapter.

In the rst chapter, we present a joint work with Chady ElMir published in the Journal of Conformal Geometry and Dynamics, cf. [START_REF] El Mir | Conformal geometric inequalities on the Klein bo le[END_REF].

In 1986, C. Bavard, cf. [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF], proved that every Riemannian Klein bo le K satis es

area(K) ≥ 2 √ 2 π sys(K) 2 .
de nitions and details about P. Pu's and C. Bla er's inequalities on the Mobius band. A selfcontained proof of the systolic volume of the Riemannian Klein bo le that do not require the use of inequalities on the Mobius band was given by C. Bavard in [START_REF] Bavard | Inégalités isosystoliques conformes pour la bouteille de Klein[END_REF]. e main tool in his proof is the method of extremal lengths, which is based also on the uniformization theorem. We refer to Section 1.2 for further details about it. is method is the main tool that we use in Chapter 1 to derive new optimal geometric inequalities on the Klein bo le relating the area to the length of loops in di erent free homotopy classes.

In order to describe these inequalities we introduce some notations. Recall that a at Klein bottle K is the quotient of C by the group generated by a glide re ection σ and a translation t. e fundamental group π 1 (K) can be identi ed with the deck group σ, t generated by σ and t with their relation σtσ -1 t = e. For every a ∈ π 1 (K), denote by γ a a loop representing the homotopy class a. We distinguish three families of free noncontractible loops on the Klein bo le K as follows.

1. F σ = {γ | γ is a loop freely homotopic to γ σ or γ σt },

2. F v = {γ | γ is a loop freely homotopic to γ t }, 3. F h = {γ | γ is a loop freely homotopic to γ σ 2 }.
Furthermore, we de ne σ (resp. v , resp. h , resp. L σ ) to be the smallest length of a noncontractible free loop in F σ (resp. F v , resp. F h , resp. F σ ∪ F h ). Observe that the extremal metric g e for the systole, cf. Figure 1, satis es σ (g e ) = h (g e ) = v (g e ) = L σ (g e ).

We summarize the main results of the rst chapter in the following theorem.

Figure 1: e double of the π 4 -tubular neighborhood of the equator of the Euclidean sphere S 2 is an extremal Klein bo le.

INTRODUCTION

eorem 0.0.1. Let β > 0. For every Riemannian metric g on the Klein bo le K of conformal type β, we have 1. area(K, g) ≥ C 1 β σ (g) v (g), 2. area(K, g) ≥ C 2 β L σ (g) v (g),

3. area 3 2 (K, g) ≥ C 3 β σ (g) v (g) h (g),
where C i β , i ∈ {1, 2, 3}, are constants depending only on the conformal type β. Moreover, these inequalities are optimal.

In fact, the inequality 1. was proved by C. Bavard, cf. [START_REF] Bavard | Une remarque sur la géométrie systolique de la bouteille de Klein[END_REF], for the values of β ∈ (0, 2 ln π 4 + a 0 2 ], where a 0 ∈ (0, π 2 ) is the unique solution of the equation tan(a 0 ) = 2a 0 . We complete his work to cover all the remaining conformal classes. Observe also that the inequality 3. involves three di erent lengths. To our knowledge, there are no other known systolic-like inequalities, even non-optimal ones, for which the number of lengths is greater than the dimension of the manifold. We refer to Section 1.1.2 and Table 1.1 of Chapter 1 for details about the corresponding extremal metrics in each conformal class.

As a consequence of eorem 0.0.1, we conclude that Corollary 0.0.2. ere does not exist any positive universal constant c such that the inequality area(K, g) ≥ c σ (g) v (g), holds for every Riemannian metric g on the Klein bo le. Similarly, there does not exist any positive universal constant c such that the inequality area 3 2 (K, g) ≥ c σ (g) v (g) h (g), holds for every Riemannian metric g on the Klein bo le.

On the other hand, we have Corollary 0.0.3. For every Riemannian metric g on the Klein bo le K,

area(K, g) > 1 2 L σ (g) v (g).
Moreover, the inequality is optimal.

In the second chapter, we present a joint work with Stéphane Sabourau, published in the Journal of Topology and Analysis, cf. [START_REF] Sabourau | Optimal systolic inequalities on Finsler Mobius bands[END_REF].

is chapter is dedicated to the study of systolic inequalities for Finsler Mobius bands. Loosely speaking, a Finsler metric F is de ned as a Riemannian metric except that its restriction to a tangent plane is no longer a Euclidean norm but a Minkowski norm. From a dynamical point of view, the function F 2 can be considered as a Lagrangian which induces a Lagrangian ow on the tangent bundle T M of M . us, Finsler manifolds can be considered as degree 2 homogeneous Lagrangian systems. e trajectories of the Lagrangian correspond to the geodesics of the Finsler metric.

ere exist several de nitions of volume for Finsler manifolds which coincide in the Riemannian case. In this chapter, we consider the Holmes-ompson volume vol HT . Recall that the Holmes-ompson volume of an n-dimensional Finsler manifold is equal to the symplectic volume of its unit co-disc bundle divided by the volume of the Euclidean unit ball of dimension n.

We refer to Section 2.2 for further details about Finsler metrics.

Since there is no Finsler analog to the uniformization theorem, studying the systolic volume of Finsler surfaces is quite delicate. e rst optimal Finsler systolic inequality has been obtained by S. Ivanov [START_REF] Ivanov | On two-dimensional minimal llings[END_REF]. He proved that the optimal systolic inequality on the projective plane for Riemannian metrics also holds for Finsler metrics. Namely, if RP 2 is a Finsler projective plane, then area(RP 2 ) ≥ 2 π sys(RP 2 ) 2 .

e extremal Finsler metrics are those all of whose geodesics are closed and of the same length.

Contrary to the Riemannian case, the Finsler systolic constant of the two-torus is the same as that of the projective plane. More precisely, S. Sabourau [START_REF] Sabourau | Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic[END_REF] proved that if T 2 is a Finsler two-torus, then area(T 2 ) ≥ 2 π sys(T 2 ) 2 .

Finsler two-tori that are homothetic to the quotient of R 2 , endowed with a parallelogram norm || . ||, by a la ice whose unit disk of || . || is a fundamental domain are extremal. However, we do not know if those are the only extremal metrics. A sharp systolic inequality for asymmetric Finsler two-tori can be found in [ABT].

e main result of the second chapter is the following. eorem 0.0.4. Let M be a Finsler Mobius band. Let λ := h(M) sys(M) . en

area(M) sys(M) h(M) ≥    2 π if λ ∈ (0, 1] 1 π λ+1 λ otherwise.
Moreover, the above inequalities are optimal for every value of λ ∈ (0, +∞).

Here, h(M) represents the minimal length of arcs with endpoints on the boundary ∂M which are not homotopically trivial relative to ∂M. We refer to Examples 2.5.4 and 2.6.2 for a description of an extremal metric family. As in the case of Finsler two-tori, we do not know if these are the only extremal metrics.

INTRODUCTION

e proof of eorem 0.0.4 is divided into two cases. e rst case is when h(M) ≥ sys(M).

Its proof proceeds as follows. We rst derive a Finsler analog of L. Keen's inequality for two-tori, cf. [START_REF] Keen | An extremal length on a torus[END_REF]. More precisely, Proposition 0.0.5. Let T 2 be a Finsler two-torus. ere exist two closed curves of lengths a and b generating the rst integral homology group of T 2 such that area(T 2 ) ≥ 2 π ab.

Equality holds if T 2 is homothetic to the quotient of R 2 , endowed with a parallelogram norm ||.||, by a la ice generated by two vectors of lengths a and b, parallel to the sides of the unit ball of ||.||.

e proof of Proposition 0.0.5 can be used as well in the Riemannian case and yields L. Keen's result without making use of the uniformization theorem as in the original proof. With the help of Proposition 0.0.5, we derive a similar result for Finsler cylinders C. More precisely, we obtain

area(C) ≥ 2 π sys(C)h(C)
where h denotes the height of the cylinder, that is, the distance between its two boundary components. Using this result and the Finsler systolic inequality on the projective plane, we obtain optimal systolic inequalities on wide Finsler Mobius bands, cf. Section 2.5.

Furthermore, in order to prove the systolic inequality for Finsler Mobius bands M that satisfy h(M) < sys(M),

we derive a systolic-height inequality which holds true whenever h(M) sys(M) is a dyadic rational in the interval (0, 1) and deduce the result by continuity of the volume, the height and the systole over Finsler metrics. Hence the systolic inequality holds for narrow Mobius bands too, cf. Section 2.6. e optimal systolic inequality for Finsler Klein bo les is still unknown. However, using eorem 0.0.4, we deduce that Finsler Klein bo les with some natural symmetries satisfy the same systolic volume as Finsler two-tori T 2 and Finsler projective planes RP 2 . In particular, we prove eorem 0.0.6. Let K be a Finsler Klein bo le with a soul, soul-switching or rotational symmetry.

en area(K) ≥ 2 π sys(K) 2 . (0.0.1)
Moreover, the inequality is optimal.

Recall that in the Riemannian case, eorem 0.0.6 follows directly from the uniformization theorem, an average argument over the isometry group, and the Cauchy-Schwartz inequality. In our proof, we do not use any of these arguments. Instead, we rely on the sharp systolic-like inequality we established for Finsler Mobius bands.

We end this chapter by proposing the following conjecture

Conjecture. Let K be a Finsler Klein bo le. en

area(K) ≥ 2 π sys(K) 2 . (0.0.2)
A Finsler Klein bo le with systolic volume equal to 2 π is described in Example 2.7.7.

e third chapter deals with critical isosystolic genus two surfaces.

Recently, M. Katz and S. Sabourau, cf. [START_REF] Katz | An optimal systolic inequality for CAT(0) metrics in genus two[END_REF], proved that there exists a piecewise at metric g 0 on the genus two surface Σ 2 which is extremal among all nonpositively curved Riemannian metrics. is metric is composed of six regular at octagons. It admits regions where only one systolic loop, i.e., a noncontractible loop of length the systole, passes through every point. Hence this metric cannot be extremal for the general systolic inequality, i.e., without restriction on the curvature. is follows from a result of E. Calabi, cf. [START_REF] Calabi | Extremal isosystolic metrics for compact surfaces, Actes de la Table Ronde de Géométrie Di érentielle[END_REF], which says that at least two systolic loops pass through every point of a systolically extremal surface. In fact, the non-extremality of the metric g 0 for the systolic inequality can also be deduced from a result of S. Sabourau, cf. [START_REF] Sabourau | Systoles des surfaces plates singulières de genre deux[END_REF], which says that no at metric with conical singularities is extremal for the systolic inequality in genus two.

Denote by 1 (g) the length of the shortest noncontractible loop on (Σ 2 , g), that is,

1 (g) = sys(g).
Recall that g 0 designates the extremal nonpositively curved genus two surface de ned in [START_REF] Katz | An optimal systolic inequality for CAT(0) metrics in genus two[END_REF].

Let Λ 1 be the subset of the free homotopy classes generated by the systolic loops of (Σ 2 , g 0 ) and their multiples:

Λ 1 = { γ k | γ = 0, g 0 (γ) = 1 (g 0 ) and k ∈ Z * }. (0.0.3)
Similarly, let 2 (g) be the length of the shortest noncontractible loop on (Σ 2 , g) which is not homotopic to a systolic loop. at is,

2 (g) = inf γ / ∈Λ 1 ∪{0} g (γ).
Denote also by

Λ 2 = { γ | γ / ∈ Λ 1 and g 0 (γ) = 2 (g 0 )} (0.0.4)
the subset of the free homotopy classes generated by the shortest noncontractible loops of (Σ 2 , g 0 )

which are not freely homotopic to systolic loops.

INTRODUCTION

We observe that the piecewise at metric g 0 de ned on the genus two surface Σ 2 may be a potential extremal metric for another systolic problem on Σ 2 . is observation follows from some geometric properties of the surface (Σ 2 , g 0 ). First, it is known that the systolic loops cover a systolically extremal surface. In our case, the 1 -loops and the 2 -loops cover the surface Σ 2 .

Here, by an i -loop we mean a loop of length i (g 0 ) in (Σ 2 , g 0 ) whose free homotopy class lies in Λ i . Second, the unit tangent vectors of these i -loops are well distributed on each tangent plane of the surface. More precisely, their convex hull forms a regular octagon on these tangent planes. In general, the convex hull of the unit tangent vectors of systolic loops of all the known systolically extremal surfaces is symmetric. Finally, an extremal surface of genus at least two tends to have at regions, cf. [START_REF] Calabi | Extremal isosystolic metrics for compact surfaces, Actes de la Table Ronde de Géométrie Di érentielle[END_REF][START_REF] Bryant | On extremals with prescribed lagrangian densities, Manifolds and geometry[END_REF].

In [START_REF] Calabi | Extremal isosystolic metrics for compact surfaces, Actes de la Table Ronde de Géométrie Di érentielle[END_REF], E. Calabi described two piecewise at genus three surfaces and conjectured that one of them is the global minimum for the systolic inequality. Later, S. Sabourau, cf. [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF], proved that these two genus three surfaces are critical for the systolic inequality with respect to slow metric variations. Of course, since the systolic volume functional is not necessarily di erentiable, an adequate notion of criticality needs to be de ned. e notion of criticality used in [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF] is the same as the one introduced in [Na96, EI00] to study the Riemannian surfaces that maximize the product of the area with the rst nonzero eigenvalue of the Laplacian. Other notions of systolically critical metrics were used in [START_REF] Balache | Sur la systole de la sphère au voisinage de la métrique standard[END_REF][START_REF] Balache | A local optimal diastolic inequality on the two-sphere[END_REF][START_REF] Sabourau | Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic[END_REF][START_REF] Paiva | Contact geometry and isosystolic inequalities[END_REF].

In this chapter we show that the metric g 0 on the genus two surface Σ 2 is critical in the sense of [START_REF] Nadirashvili | Berger's isoperimetric problem and minimal immersions of surfaces[END_REF][START_REF] El Sou | Riemannian manifolds admi ing isometric immersions by their rst eigenfunctions[END_REF][START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF] for slow metric variations. More precisely, we prove eorem 0.0.7. Let g t be a slow metric variation of g 0 de ned on the genus two surface Σ 2 . If

1 (g t ) ≥ 1 (g 0 ) and 2 (g t ) ≥ 2 (g 0 ), then area(g t ) ≥ area(g 0 ) + o(t).
e slow metric variations involved in eorem 3.1.1 are analogous to those de ned by S. Sabourau in [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF]. We refer to De nition 3.4.1 for a precise de nition of these metric variations and to the last section of [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF] for examples. Observe for instance that deforming the regular octagons composing the extremal surface (Σ 2 , g 0 ) into non-regular octagons gives rise to a slow metric variation.

To prove our result, we rely on recent calibrating methods, cf. [START_REF] Ivanov | On two-dimensional minimal llings[END_REF][START_REF] Ivanov | Filling minimality of Finslerian 2-discs[END_REF][START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF], that do not require the uniformization theorem classically used to establish sharp systolic inequalities on surfaces. Brie y, we embed the universal cover Σ 2 of the genus two surface Σ 2 in an in nitedimensional space R ∞ as follows. Given that i (g) ≥ i (g 0 ), we construct a 1-Lipschitz equivariant map

Ψ g : Σ 2 → R ∞
using the Busemann functions induced by the li s of the i -loops of Σ 2 , cf. Section 3.3. en, we introduce an appropriate in nitesimally calibrating two-form ω on R ∞ whose pull-back passes to the quotient on the surface Σ 2 . is allows us to show that

Σ 2 (Ψ g ) * ω ≤ c 0 area(Σ 2 , g)
for every Riemannian metric g on Σ 2 , where c 0 is a sharp positive constant. Moreover, the equality holds if g = g 0 . Finally, we prove that if (g t ) t≥0 is a slow metric variation then

Σ 2 (Ψ gt ) * ω - Σ 2 (Ψ g 0 ) * ω = o(t).
is completes the proof, cf. Section 3.4.

INTRODUCTION

Chapter 1

Conformal geometric inequalities on the Klein bottle

Nous démontrons trois inégalités géométriques optimales conformes sur la bouteille de Klein reliant l'aire au produit des longueurs des plus courts lacets noncontractiles dans des classes d'homotopie libres di érentes. Pour chaque classe conforme, nous décrivons la métrique extrémale réalisant le cas d'égalité.

Abstract. We obtain three optimal conformal geometric inequalities on Riemannian Klein bo les relating the area to the product of the lengths of the shortest noncontractible loops in di erent free homotopy classes. We describe the extremal metrics in each conformal class.

Introduction, preliminaries and results

Among all Riemannian metrics on a given compact di erentiable manifold, the most interesting ones are those that extremize some Riemannian invariant. An interesting problem, for example, is to study metrics which maximize the ratio sys(M,g) n vol(M,g) over the set of all Riemannian metrics g on a given n-dimensional di erentiable manifold M , where sys(M, g) denotes the systole of (M, g), i.e., the least length of a non-contractible closed curve. Concerning this problem, called isosystolic problem, it is known that the extremal metric for the 2-dimensional torus is the at hexagonal metric (unique up to a homothety). It is due to C. Loewner in 1949 (unpublished). His student, P.M. Pu, showed in 1952 that the extremal metric on the projective plane is the spherical metric, cf. [START_REF] Pu | Some inequalities in certain non-orientable Riemannian manifolds[END_REF]). Nevertheless, as in many works related to the isosystolic problem, various constraints can be put on the set of Riemannian metrics under consideration. For example, we may restrict ourselves to the set of nonpositive curvature metrics cf.

σ (g) v (g) ≤ C β area(g) (1.1.1)
where C β is a positive constant that depends only on the conformal type β of g. Note that the optimal constant C β is not bounded from above over the set of conformal types β. Also, in the same article, Bla er proved the following conformal lower bound on the area of M

sys(g) v (g) ≤ C β area(g) (1.1.2)
where C β is a positive constant that depends only on the conformal type β of g. Contrarily to the previous case, the optimal constant C β is bounded from above over the set of conformal types β.

More precisely, sup

β C β = 2.
Hence, the optimal inequality sys(g) v (g) ≤ 2 area(g)

14CHAPTER 1: CONFORMAL GEOMETRIC INEQUALITIES ON THE KLEIN BOTTLE holds for every Riemannian metric g on M.

Variants of C. Bla er's problem were studied by L. Keen in [START_REF] Keen | An extremal length on a torus[END_REF] and J. Hebda in [START_REF] Hebda | Two Geometric Inequalities for the Torus[END_REF] on the 2-dimensional torus. Keen obtained a lower bound on the area in terms of the product of the least lengths of two loops generating the fundamental group. On the other hand, Hebda obtained a lower bound on the square of the area in terms of the three primitive length spectrum. e equality in both cases is a ained by the at hexagonal metric.

In this chapter, we prove three types of inequalities on the Klein bo le K in the same spirit as Bla er's inequalities. e classes of curves we consider are (free) homotopy classes of loops that are candidates to realize the systole. In particular, the role of the family F will be taken by a free homotopy class of loops representing the vertical translation in π 1 (K). e main tool in our proof is the method of extremal length which was used by C. Bavard in 1988 to prove conformal isosystolic inequalities on the Klein bo le cf. [START_REF] Bavard | Inégalités isosystoliques conformes pour la bouteille de Klein[END_REF]. It can be applied to the Mobius band with boundary and provides an alternative proof of Bla er's inequalities (1.1.1) and (1.1.2).

e Klein bottle

A at Klein bo le is the quotient of C by the group generated by the maps σ : z → z + π and t v : z → z + 2iβ. e induced at metric on the Klein bo le will be denoted by g β . By the uniformization theorem, every Riemannian metric g on the Klein bo le is conformally equivalent to a unique at metric g β for some β ∈ (0, +∞). e parameter β represents the conformal type of the metric g. We call t h := σ 2 a horizontal translation and identify the fundamental group π 1 (K) with the deck group σ, t v . Note that any Klein bo le K is obtained by gluing two Mobius bands M 1 and M 2 along their boundaries. With the previous notations, fundamental domains

of K, M 1 and M 2 , are [-π 2 , π 2 ] × [-β, β], [-π 2 , π 2 ] × [ -β 2 , β 2 ] and [-π 2 , π 2 ] × [-β, -β 2 ] ∪ [ β 2 , β] respectively.
By C. Bavard's theorem, cf. [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF], every Riemannian Klein bo le K satis es the isosystolic inequality

sys(K) 2 ≤ π 2 √ 2 area(K)
where the equality is a ained by a spherical metric outside a singular line (see [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF], [START_REF] Sakai | A proof of the isosystolic inequality for the Klein bo le[END_REF] and [START_REF] El Mir | Sur la géométrie systolique des variétés de Bieberbach[END_REF] p. 100 for a detailed description of the extremal Klein bo le). For details and (many) open problems in systolic geometry see the book of Katz [START_REF] Katz | Systolic geometry and topology[END_REF] and the paper of M. Gromov [START_REF] Gromov | Filling Riemannian manifolds[END_REF].

In this chapter, we consider the following distinct families of homotopy classes of closed curves on the Klein bo le K as follows.

De nition 1.1.1. For every a ∈ π 1 (K), denote by γ a a loop representing the homotopy class a. en we de ne

1. F σ = {γ | γ is a loop freely homotopic to γ σ or γ σt }, 2. F v = {γ | γ is a loop freely homotopic to γ tv }, 3. F h = {γ | γ is a loop freely homotopic to γ σ 2 }.
Given a Riemannian metric on K, the least length of a closed curve in the rst (resp. second, resp. third) family will be denoted by σ (resp. v , resp. h ). Also, denote by L σ the least length of a loop in F σ ∪ F h .

Remark 1.1.2. An extremal metric g ext for the isosystolic inequality on the Klein bo le K satis es

σ (g ext ) = L σ (g ext ) = v (g ext ).
Its conformal type β is equal to 2 ln(tan( 3π 8 )), cf. [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF].

Finally, denote by a 0 the unique real x ∈ (0, π 2 ) such that tan(x) = 2x (a 0 ≈ 1.1655).

Two families of Riemannian metrics on the Klein bottle

We denote by C 0,∞ (R) the set of continuous and piecewise smooth functions on R. Let h β be a Riemannian metric in C R 2 , periodic with respect to the fundamental domain [-π 2 , π 2 ] × [-β, β] and satisfying

(h β ) (x,y) = ϕ(y)(dx 2 + dy 2 ), (1.1.3)
where ϕ ∈ C 0,∞ (R) is a positive, even and 2β-periodic function. en there exists a di eomorphism G : C → C for which the Riemannian metric h b = (G -1 ) * h β satis es

(h b ) (u,v) = f 2 (v)du 2 + dv 2 , (1.1.4)
where f ∈ C 0,∞ (R) is positive, even and 2b-periodic function. To see this, let φ(y) = y 0 ϕ(t)dt and f = (ϕ • φ -1 ) 1 2 and de ne the map G by G(x, y) = (x, φ(y)).

(1.1.5) e Riemannian metric h b is then periodic with respect to the fundamental domain

[-π 2 , π 2 ] × [-2b, 2b
]. e quotient of (C, h b ) by the subgroup of isometries < σ, t v >, where σ : z → z + π and t v : z → z + 4ib, is a Riemannian Klein bo le. Its conformal type is

β = 2b 0 dt f (t)
.

Now, we will introduce two families of metrics on C. ey induce families of metrics on the Klein bo le that will be used later in this chapter as conformally extremal metrics for three types of inequalities on the Klein bo le. For every b ∈ (0, +∞) and ω ∈ (0, b], they are as follows.

1. the metric S b on C periodic with respect to the fundamental domain [-π 2 , π 2 ] × [-2b, 2b] and de ned by

(S b ) (u,v) = f 2 b (v)du 2 + dv 2
where f b is the unique one-variable function invariant by the translation (u, v) → (u, v + 2b) which agrees with cosine on [-b, b]. It is spherical outside the singular lines v = nb, where n ∈ Z.

2. the metric SF b,ω on C periodic with respect to the fundamental domain

[-π 2 , π 2 ]×[-2b, 2b
] and de ned by

(SF b,ω ) (u,v) = f 2 b,ω (v)du 2 + dv 2
where f b,ω is the unique one-variable function invariant by the translation (u, v) → (u, v+ 2b) which agrees with cosine on [-ω, ω] and equal to the constant cos(ω) on [ω, 2b -ω],

cf. [Ba88, Fig. 1]. It is spherical on the band R × [-ω, ω] and its images by the translations (u, v) → (u, v + 2nb), where n ∈ Z, and at elsewhere.

We will denote by (K, S b ) the Riemannian Klein bo le obtained by taking the quotient of (C, S b ) by the subgroup < σ, t v >. Its conformal type is

β = 2 ln(tan( π 4 + b 2 )).
Similarly, (K, SF b,ω ) will denote the Riemannian Klein bo le obtained by taking the quotient of (C, SF b,ω ) by the subgroup < σ, t v >. Its conformal type is

β = 2 ln(tan( π 4 + ω 2 )) + 1 cos(ω) (2b -2ω).

Geometric inequality of type σ v

Our rst result studies optimal conformal inequalities of the form

σ v ≤ C β area
where C β is a constant depending only on the conformal type β. We call such a relation a " geometric inequality of type σ v ". Actually, it extends the following result of C. Bavard to all the conformal classes of the Klein bo le.

eorem 1.1.3 C. Bavard [START_REF] Bavard | Une remarque sur la géométrie systolique de la bouteille de Klein[END_REF]. Let a 0 ∈ (0, π 2 ) be such that tan(a 0 ) = 2a 0 and 0 < β ≤ 2 ln tan π 4 + a 0

2

. en, for every Riemannian metric g on the Klein bo le K of conformal type β,

we have the following optimal inequality

σ (g) v (g) ≤ arcsin e β -1 e β +1 e β -1 e β +1 area(g).
e equality is a ained if and only if g is proportional to the spherical metric S b for b satisfying β = 2 ln(tan( π 4 + b 2 )).

We complete C. Bavard's study by providing an optimal inequality for the remaining conformal classes.

eorem 1.1.4. Let a 0 ∈ (0, π 2 ) be such that tan(a 0 ) = 2a 0 and β > 2 ln tan π 4 + a 0

2

. Let

ω 1 ∈ [a 0 , π
2 ) be de ned by the equation

2 sin(ω 1 ) = β -2 ln tan π 4 + ω 1 2 cos 2 (ω 1 ) + 4ω 1 cos(ω 1 ).
en, for every Riemannian metric g on the Klein bo le K of conformal type β, we have the following optimal inequality

σ (g) v (g) ≤ 1 2 cos(ω 1 ) area(g).
Moreover, the equality is a ained if and only if g is proportional to the spherical-at metric SF b,ω 1 for b = tan(ω 1 ) -ω 1 .

Remark 1.1.5. e bound β ≤ 2 ln tan π 4 + a 0 2 found by C. Bavard in eorem 1.1.3 is actually the critical value of the conformal type for the transition in the shape of the extremal metrics from spherical to spherical-at.

Corollary 1.1.6. ere does not exist any ( nite) positive universal constant c such that the inequality σ (g) v (g) ≤ c area(g) holds for every Riemannian metric g on the Klein bo le.

Geometric inequality of type L σ v

e second part of this chapter is devoted to establishing optimal conformal inequalities of the form

L σ v ≤ C β area.
Note that in the case of the Mobius band M, L σ is just the systole of M. We call such a relation a "geometric inequality of type L σ v ".

eorem 1.1.7. Let β > 0. For every Riemannian metric g on the Klein bo le K of conformal type β, we have

L σ (g) v (g) ≤ C β area(g)
where

C β =    e β +1 e β -1 arcsin e β-1 e β +1 if 0 < β ≤ 2 ln 2 + √ 3 2 3 • 3β+4π-6 ln(2+ √ 3) 4 √ 3+β-2 ln(2+ √ 3) if β > 2 ln 2 + √ 3 . 18CHAPTER 1: CONFORMAL GEOMETRIC INEQUALITIES ON THE KLEIN BOTTLE
Moreover, the equality is a ained if and only if g is proportional to the spherical metric S b , for b satisfying β = 2 ln(tan( π 4 + b 2 )), in the rst case and to the spherical-at metric SF b, π 3 , for b satisfying β = 2 ln(2 + √ 3) + 4(b -π 3 ), in the second case.

Since the supremum of the conformal constant C β over β is equal to 2, we have Corollary 1.1.8. For every Riemannian metric g on the Klein bo le K, we have

L σ (g) v (g) < 2 area(g).
e inequality is optimal.

Geometric inequality of type σ v h

In the third part, we establish optimal conformal inequalities of the form

σ v h ≤ C β area 3 2
which we call a "geometric inequality of type σ v h ". To our knowledge, this kind of inequality involving the product of three lengths has never been considered before. It is distinguished by the fact that the area to the power 3 2 is bounded by the product of all the natural candidates for the systole eorem 1.1.9. Let β > 0 and let ω 2 ∈ (0, π 2 ) be de ned by the equation

β = 2 ln(tan( π 4 + ω 2 2 )) + 2 cos(ω 2 ) tan(ω 2 ) -ω 2 + tan 2 (ω 2 ) -ω 2 tan(ω 2 ) + ω 2 2 .
en, for every Riemannian metric g on the Klein bo le K of conformal type β, we have the following optimal inequality

σ (g) v (g) h (g) ≤ C β area(g) 3 2
where

C β = √ π 3 √ 3 • b 4 -4bω 2 + ω 2 2 + ω 2 4 -2b 2 (-2 + ω 2 2 ) 1 4 (2b -ω 2 ) (b -ω 2 ) (b -ω 2 )b with b = tan(ω 2 ) + tan 2 (ω 2 ) -ω 2 tan(ω 2 ) + ω 2 2 .
Moreover, the equality is a ained if and only if g is proportional to the spherical-at metric SF b,ω 2 .

Now, if we replace b by tan(ω 2 ) + tan 2 (ω 2 ) -ω 2 tan(ω 2 ) + ω 2 2 in C(ω 2 ) = b 4 -4bω 2 + ω 2 2 + ω 2 4 -2b 2 (-2 + ω 2 2 ) 1 4 (2b -ω 2 ) (b -ω 2 ) (b -ω 2 )b
we obtain a continuous increasing function

C :] 2 π , +∞[→] 2 √ π 3 √
3 , +∞[ which tends to in nity as ω 2 → 0 (i.e. when b → 0). erefore, we derive Corollary 1.1.10. ere does not exist any ( nite) universal constant c such that the inequality

σ (g) v (g) h (g) ≤ c area(g) 3 2
holds for every Riemannian metric g on the Klein bo le K.

Remark 1.1.11. Unlike Corollaries 1.1.6 and 1.1.10, Corollary 1.1.8 provides a uniform upper bound on the Riemannian ratio Lσ(g) v (g) area(g) .

e main tools for the proofs

e key tool in our proofs is the method of extremal length initiated by B. Fuglede in [Fu57], J.

A. Jenkins in [START_REF] Jenkins | Univalent functions and conformal mapping[END_REF] and M. Gromov in [START_REF] Gromov | Filling Riemannian manifolds[END_REF]. It was used later by C. Bavard in the se ing of isosystolic geometry, cf. [START_REF] Bavard | Inégalités isosystoliques conformes pour la bouteille de Klein[END_REF][START_REF] Bavard | Inégalités isosystoliques conformes[END_REF]. is method charaterizes a conformally extremal Riemannian manifold by means of its closed geodesics. See e.g. L. V. Ahlfors' book [Al73] and Rodin's paper [START_REF] Rodin | Extremal length and geometric inequalities[END_REF] and the references therein for more details and further applications of the method of extremal length.

A maximality criterion

Let (M, g) be a closed Riemannian manifold and Γ be a family of recti able curves on M . For every Radon measure µ on Γ, we associate a measure * µ on M by se ing, for ϕ

∈ C 0 (M, R), * µ, ϕ = µ, ϕ
where ϕ(γ) = ϕ • γ(s)ds and ds is the arc-length of γ with respect to g.

eorem 1.2.1. ([Ba88],[Ba92] and [Je65]) Let M be a closed manifold. Let S i , where i ∈ {1, • • • , p},
be some families of recti able curves on M such that S i ∩ S j = ∅ for all i = j. Denote by i (g)

the least length of a curve in S i with respect to a Riemannian metric g on M . Suppose that g e is a

Riemannian metric on M for which there exists a positive measure µ on Γ = S 1 ∪• • •∪S p satisfying the following three conditions:

1. for each i ∈ {1, • • • , p}, all the curves in S i have the same length with respect to g e

2. m 1 1 (g e ) = m 2 2 (g e ) = • • • = m p p (g e ) 3. * µ = dg e
where m i is the mass of the measure µ on S i and dg e is the volume measure of (M, g e ). en, for every Riemannian metric g on M conformal to g e , we have

1 (g) • • • p (g) vol p 2 (g) ≤ 1 (g e ) • • • p (g e ) vol p 2 (g e )
.

(1.2.1)

Furthermore, the equality holds if and only if g is homothetic to g e .

Since our version of eorem 1.2.1 is slightly more general than in the aforementioned references (as it holds for an arbitrary number of curve families S i ), we present a proof of it. It is straightforward by the following lemma.

Lemma 1.2.2. Let S i , where i ∈ {1, • • • , p}, be families of recti able curves on a given closed manifold M such that S i ∩ S j = ∅ for all i = j. Denote by i (g) the least length of a curve in S i with respect to a Riemannian metric g on M . Let g e be a Riemannian metric on M such that there exists a positive measure µ on Γ = S 1 ∪ • • • ∪ S p satisfying the three conditions of eorem 1.2.1.

en,

1 (g) • • • p (g) ≤ vol p 2 (g e ) p p m 1 • • • m p vol p 2 (g) (1.2.2)
where m i is the mass of the measure µ on S i . Furthermore, the equality holds if and only if g is homothetic to g e .

Proof. Let g be a Riemannian metric conformal to g e , that is, g = φ 2 g e . We have

m 1 1 (g) + • • • + m p p (g) ≤ S 1 φ(γ)dµ(γ) + • • • + Sp φ(γ)dµ(γ) (1.2.3) = Γ φ(γ)dµ(γ) = M φ(x)d( * µ)(x) = M φ(x)dg e ≤ M φ 2 (x)dg e 1 2 M dg e 1 2 (1.2.4) = vol(g)vol(g e ).
When g is equal to g e , equality in (1.2.3) is a ained because g e satis es condition (1). Moreover, it is straightforward that equality in (1.2.4) is a ained if and only if g is equal to g e by Cauchy-Schwartz. Using the arithmetic and geometric means inequality, we derive

p. m 1 • • • m p 1 (g) • • • p (g) 1 p ≤ m 1 1 (g) + • • • + m p p (g)
with equality if and only if

m 1 1 (g) = m 2 2 (g) = • • • = m p p (g).
Finally, combining the two inequalities, we obtain that, under the required conditions, the following inequality holds

p p . m 1 • • • m p . 1 (g) • • • p (g) ≤ vol(g)vol(g b ) p 2 ,
with equality if and only if g is equal to g e .

us, proving the conformal extremality of a metric requires to nd disjoint families of recti able curves S 1 , • • • , S n on M and de ne a measure on them that satis es the three conditions of the previous theorem.

Setting the curve families

To prove our results, we will make use of three families of curves in (K, SF b,ω ).

De nition 1.2.3. Let (K, SF b,ω ) be the spherical-at Klein bo le de ned in Section 1.1.2. De ne three families of loops as follows.

1. For each θ ∈ R/πZ and each a ∈

[-ω, ω] ∪ [2b -ω, 2b] ∪ [-2b, -2b + ω],
the loop α a θ is the geodesic (image in (K, SF b,ω ) of a great circle) going through the points (θ -π/2, 0), (θ, a). Let

S 1 (ω) = {α a θ | |a| ≤ ω or 2b -ω ≤ |a| ≤ 2b, θ ∈ R/πZ} . 2. For each u ∈ R/πZ, γ u is the vertical loop de ned by γ u (t) = (u, t) with |t| ≤ 2b. Let S 2 = {γ u | u ∈ R/πZ} . 3. For each v ∈ R satisfying ω ≤ |v| ≤ 2b -ω, δ v is the horizontal loop de ned by δ v (t) = (t, |v|) with |t| ≤ π 2 . Let S 3 (ω) = {δ v | ω ≤ |v| ≤ 2b -ω} .
Remark 1.2.4. For each i ∈ {1, 2, 3}, the curves in S i have the same length with respect to the metric SF b,ω . Also note that a curve in S 1 , resp. S 2 , resp. S 3 , belongs to the family F σ , resp. F v , resp. F h .

We move now to the next step, i.e., verifying the three conditions of eorem 1.2.1. Conditions

(1) and (2) are easy to verify while Condition (3) requires technical (but not simple) methods of calculation. In the following we will consider a subset of the family of curves S 1 (ω) in (K, SF b,ω ) and equip it with a measure µ depending on a function h of one parameter (since SF b,ω has an isometry group of dimension 1). en, we will calculate the measure * µ in terms of the volume measure of SF b,ω .

e calculation of * µ on a spherical region of K

Let b ∈ (0, +∞) and ω ∈ (0, b]. We consider on (K, SF b,ω ) the family of curves S 1 (ω) ⊂ S 1 (ω)

de ned by S 1 (ω) = {α a θ ⊂ (K, SF b,ω ) | -ω ≤ a ≤ ω, θ ∈ R/πZ}
where α a θ is the great circle introduced in De nition 1.2.3.

Lemma 1.2.5. Let µ be a measure on S 1 (ω) de ned by

dµ(α a θ ) = h(a) da dθ if a ≥ 0 h(-a) da dθ if a < 0 where h : [0, ω] → R is a continuous function. en, we have * µ = 2χ {|v|≤ω} ω |v| cos 2 (v) -cos 2 (a) -1 2 h(a) da d(SF b,ω )
where d(SF b,ω ) is the volume measure of SF b,ω .

Proof. First, we compute the equation of α a θ . Suppose α a θ (t) = (u(t), v(t)). en, from the

classical geodesic equation d 2 x i dt 2 + j,k Γ i j,k dx k dt dx j
dt = 0 (see for example [START_REF] Gallot | Riemannian Geometry[END_REF] p. 81), where

Γ i j,k are the Christo el symbols, x 1 = u and x 2 = v, we derive u -2 tan(v)u v = 0 and v + sin(v) cos(v)u 2 = 0. is shows that du dv = c cos(v) cos 2 (v) -c 2 (1.2.5)
where c is a constant. e solution of the di erential equation

(1.2.5) is sin(u -d) = c tan(v),
where d is a constant. Using the fact that α a θ goes through (θ -π/2, 0) and (θ, a), we obtain

sin(u -θ + π 2 ) = 1 tan(a) tan(v). (1.2.6) Equation (1.2.6) shows that we can write α a θ (u) = (u + θ, v (u, a)), with v verifying v(u, -a) = -v(u, a). Now, let φ ∈ C 0 (K, R).
en, by the de nition of * µ, we have

K φ(u, v) d( * µ)(u, v) = S 1 (ω) φ(α a θ ) dµ(α a θ )
where

φ(α a θ ) = α a θ φ (α a θ (s)) ds = π 2 -π 2 φ(u + θ, v(u, a)) cos 2 (v(u, a)) + ∂v ∂u (u, a) 2 du. en, K φ(u, v) d( * µ)(u, v) = π 2 -π 2 0 -ω φ(α a θ )h(-a) da dθ + π 2 -π 2 ω 0 φ(α a θ )h(a) da dθ = π 2 -π 2 ω 0 0 -π 2 φ(u + θ, -v(u, a)) cos 2 (v(u, a)) + ∂v ∂u (u, a) 2 h(a) du da dθ + π 2 -π 2 ω 0 π 2 0 φ(u + θ, -v(u, a)) cos 2 (v(u, a)) + ∂v ∂u (u, a) 2 h(a) du da dθ + π 2 -π 2 ω 0 0 -π 2 φ(u + θ, v(u, a)) cos 2 (v(u, a)) + ∂v ∂u (u, a) 2 h(a) du da dθ 24CHAPTER 1: CONFORMAL GEOMETRIC INEQUALITIES ON THE KLEIN BOTTLE + π 2 -π 2 ω 0 π 2 0 φ(u + θ, v(u, a)) cos 2 (v(u, a)) + ∂v ∂u (u, a) 2 h(a) du da dθ
Next, we apply for the rst two integrals in the previous expression the change of variables y → -v(u, a), whose Jacobian is equal to -∂v ∂u (u, a), and we write k 1 (y, a) = u. For the last two integrals, we make use of the change of variables y → v(u, a), whose Jacobian is equal to ∂v ∂u (u, a), and we write k 2 (y, a) = u. In either case, let z(y, a) = ∂v ∂u (u, a). We derive

K φ(u, v) d( * µ)(u, v) = 2 π 2 -π 2 ω 0 0 -a φ(k 1 (y, a) + θ, y) cos 2 (y) + z 2 (y, a) h(a) |z(y, a)| dy da dθ + 2 π 2 -π 2 ω 0 a 0 φ(k 2 (y, a) + θ, y) cos 2 (y) + z 2 (y, a) h(a) |z(y, a)| dy da dθ.
Next, in the previous expression, we make the change of variables x → k 1 (y, a) + θ for the rst integral and x → k 2 (y, a) + θ for the second integral. e Jacobian for both changes of variables is equal to 1. We obtain

K φ(u, v) d( * µ)(u, v) = 2 π 2 -π 2 ω 0 0 -a φ(x, y) cos 2 (y) + z 2 (y, a) h(a) |z(y, a)| dy da dx + 2 π 2 -π 2 ω 0 a 0 φ(x, y) cos 2 (y) + z 2 (y, a) h(a) |z(y, a)| dy da dx = 2 π 2 -π 2 0 -ω ω -y φ(x, y) cos 2 (y) + z 2 (y, a) h(a) |z(y, a)| da dy dx + 2 π 2 -π 2 ω 0 ω y φ(x, y) cos 2 (y) + z 2 (y, a) h(a) |z(y, a)| da dy dx = 2 π 2 -π 2 ω -ω φ(x, y) ω |y| cos(y) z(y, a) 2 + 1 h(a) da dy dx . Hence, we derive * µ = 2 χ {|v|≤ω} cos(v) ω |v| cos(v) z(v, a) 2 + 1 h(a)da d(SF b,ω )
where d(SF b,ω ) is the volume element of (K, SF b,ω ). From equation (1.2.5), we obtain

z 2 (v, a) = cos 2 (v) cos 2 (a) cos 2 (v) -cos 2 (a)
.

is shows that * µ = 2χ {|v|≤ω} ω |v| cos 2 (v) -cos 2 (a) -1 2 h(a)da d(SF b,ω ).

Proof of the geometric inequality of type σ v

In the following, we prove the geometric inequality of type σ v for β > 2 ln tan π 4 + a 0 2 .

As in the case β ≤ 2 ln tan π 4 + a 0 2 , we will consider both curve families S 1 (ω 1 ) and S 2 , cf.

[Ba06, Corollary 2]. Proof of eorem 1.1.4. Let β > 2 ln tan π 4 + a 0 2 . ere exists a unique ω 1 ∈ [a 0 , π 2 ) such that 2 sin(ω 1 ) = β -2 ln tan π 4 + ω 1 2 cos 2 (ω 1 ) + 4ω 1 cos(ω 1 ). (1.3.1)
We endow K with the metric SF b,ω 1 de ned in Section 1.1.2, where

b = tan(ω 1 ) -ω 1 . (1.3.2)
Consider the two families of curves S 1 (ω 1 ) and S 2 , cf. Section 1.2.2. Each curve in S 1 (ω 1 ) (resp.

S 2 ) has length equal to π (resp. 4b) with respect to SF b,ω 1 . Hence, condition (1) of eorem 1.2.1 is satis ed. Now, let

h : [-ω 1 , ω 1 ] -→ R a → sin(|a|) π cos(a)
cos 2 (a) -cos 2 (ω 1 )

We de ne on the family S 1 (ω 1 ) the measure

µ 1 = h(a) da ⊗ dθ where h(a) =      h(a) if a ∈ [-ω 1 , ω 1 ] h(a -2b) if a ∈ [2b -ω 1 , 2b] h(a + 2b) if a ∈ [-2b, -2b + ω 1 ]
On the family S 2 , we de ne the measure

µ 2 = cos(ω 1 )du.
e mass m 1 of the measure µ 1 on S 1 (ω 1 ) is equal to 4 sin(ω 1 ) -4ω 1 cos(ω 1 ) and the mass m 2 of the measure µ 2 on S 2 is equal to π cos(ω 1 ). Now, since σ (SF b,ω 1 ) = π and v (SF b,ω 1 ) = 4b, condition (2) of eorem 1.2.1 is satis ed. Next, it can be easily veri ed that * µ 2 = cos(ω 1 )

f b,ω 1 (v) d(SF b,ω 1 )
where f b,ω 1 is the unique one-variable function invariant by the translation (u, v) → (u, v + 2b)

which agrees with cosine on [-ω 1 , ω 1 ] and equal to the constant cos(ω 1 ) on [ω 1 , 2b -ω 1 ] (see Section 1.1.2). Moreover, by Lemma 1.2.5, we have

χ {|v|≤ω 1 } * µ 1 = 2χ {|v|≤ω 1 } ω 1 |v| cos 2 (v) -cos 2 (a) -1 2 h(a) da d(SF b,ω 1 )
en, we derive

χ {|v|≤ω 1 } * µ 1 = χ {|v|≤ω 1 } 1 - cos(ω 1 ) f b,ω 1 (v) d(SF b,ω 1 ).
Finally, since f b,ω 1 and h are invariant by the translation

(u, v) → (u, v + 2b), we have * µ 1 = 1 - cos(ω 1 ) f b,ω 1 (v) d(SF b,ω 1 ).
Hence, we get

* µ 1 + * µ 2 = d(SF b,ω 1 ).
is is condition (3) of eorem 1.2.1. Finally, from eorem 1.2.1 and inequality (1.2.2), we derive

σ (g) v (g) ≤ 1 2 cos(ω 1 ) area(g)
which holds for every Riemannian metric g conformal to SF b,ω 1 . e equality is a ained if and only if g is homothetic to SF b,ω 1 , and the result follows.

Recall that M denotes the Mobius band with boundary obtained by taking the quotient of R × [-β, β] by the group generated by the map σ : z → z+π. We denote by g β the at metric induced by such quotient. e parameter β represents the conformal type of any metric conformal to g β .

We recover Bla er's result for the Mobius Band, cf. [Bl61, eorem 2], by a simpler method.

Indeed, consider in Bavard's proof of eorem 1.1.3 and in our previous proof the restriction of S b and SF b,ω 1 on the set

E b = {(u, v) ∈ C | |v| ≤ b}. Corollary 1.3.1. [Bl61, Satz 2] Let β > 0 and a 0 ∈ (0, π 2 ) such that tan(a 0 ) = 2a 0 . Let ω 1 ∈ [a 0 , π
2 ) be de ned by the equation

sin(ω 1 ) = β -ln tan π 4 + ω 1 2 cos 2 (ω 1 ) + 2ω 1 cos(ω 1 ).
en, for every Riemannian metric g on the Mobius band M of conformal type β, we have

σ (g) v (g) ≤ C β area(g)
where

C β =    e 2β +1 e 2β -1 arcsin e 2β -1 e 2β +1 if 0 < β ≤ ln π 4 + a 0 2 1 2 cos(ω 1 ) if β > ln π 4 + a 0 2 .
Moreover, the equality is a ained if and only if g is proportional to the spherical metric

S b restricted to E b = {(u, v) ∈ C | |v| ≤ b}, for b satisfying β = 2 ln(tan( π 4 + b 2 )
), in the rst case, and to the spherical-at metric SF b,ω 1 restricted to

E b = {(u, v) ∈ C | |v| ≤ b}, for b = tan(ω 1 ) -ω 1 , in the second case.

Proof of the geometric inequality of type L σ v

In the following, we prove the geometric inequality of type L σ v on the Klein bo le. e curve families we consider this time are S 1 ( π 3 ), S 2 and S 3 ( π 3 ).

Proof of eorem 1.1.7. e inequality in the rst case can be deduced from eorem 1.1.3 since

when 0 < β ≤ 2 ln 2 + √ 3 , i.e. when b ≤ π 3 , we have L σ (S b ) = σ (S b ). Now, since the conformal type of (K, S b ) is β = 2 ln(tan( π 4 + b 2 )), we deduce that for 0 < β ≤ 2 ln 2 + √ 3 , L σ (g) v (g) ≤ arcsin e β -1 e β +1 e β -1 e β +1
area(g).

Note that when b becomes greater than π 3 , the horizontal geodesic loops closed by the horizontal translation t h (corresponding to the singularity line of S b ) become shorter than the curves α a θ and therefore L σ is a ained by such lines.

To prove the inequality in the second case, we x β > 2 ln 2 + √ 3 and let b such that β = 2 ln(2 + √ 3) + 4(b -π 3 ) (we have b ≥ π 3 ). en we endow K with the metric SF b, π 3 . We consider the two families of curves S = S 1 ( π 3 ) ∪ S 3 ( π 3 ) and S 2 , cf. Section 1.2.2. Each curve in S, (resp. S 2 ) has length equal to π, (resp. 4b) with respect to the metric SF b, π 3 . Hence, condition (1) of eorem 1.2.1 is satis ed. Now, let

k : [-π 3 , π 3 ] -→ R a → tan(|a|) 24πb . 24b cos 2 (a)-3 √ 3-3b+π cos 2 (a)-1 4
We de ne on the family S the measure

µ 1 = k(a) da ⊗ dθ + -3 √ 3 + 3b + π 6b dv.
where

k(a) =      k(a) if a ∈ [-π 3 , π 3 ] k(a -2b) if a ∈ [2b -π 3 , 2b] k(a + 2b) if a ∈ [-2b, -2b + π 3 ] 28CHAPTER 1: CONFORMAL GEOMETRIC INEQUALITIES ON THE KLEIN BOTTLE
On the family S 2 , we de ne the measure

µ 2 = 3 √ 3 + 3b -π 12b du.
Note that these measures are positive since b ≥ π 3 . e mass m 1 of the measure µ 1 on S is equal to √ 3 + 3b -π 3 , and the mass m 2 of the measure µ 2 on S 2 is equal to 3

√ 3+3bπ-π 2 12b
. Now, since

L σ (SF b, π 3 ) = π and v (SF b, π 3 ) = 4b, condition (2) of eorem 1.2.1 is satis ed. Next, it can be easily veri ed that * µ 2 = 3 √ 3 + 3b -π 12bf b, π 3 (v) d(SF b, π 3 )
where f b, π 3 is the unique one-variable function invariant by the translation (u, v) → (u, v + 2b) which agrees with cosine on [-π 3 , π 3 ] and equal to the constant 1 2 on [ π 3 , 2b -π 3 ]. Moreover, by Lemma 1.2.5 we obtain

χ {|v|≤ π 3 } * µ 1 = 2χ {|v|≤ π 3 } π 3 |v| cos 2 (v) -cos 2 (a) -1 2 k(a) da d(SF b, π 3 )
en, we derive

χ {|v|≤ π 3 } * µ 1 = χ {|v|≤ π 3 } 1 - 3 √ 3 + 3b -π 12bf b, π 3 (v) d(SF b, π 3 ).
Finally, since f b, π 3 and k are invariant by the translation (u, v) → (u, v + 2b), we have

* µ 1 =    1 -3 √ 3+3b-π 12bf b, π 3 (v) d(SF b, π 3 ) if |v| ≤ π 3 or 2b -π 3 ≤ |v| ≤ 2b -3 √ 3+3b+π 6b d(SF b, π 3 ) if π 3 ≤ |v| ≤ 2b -π 3
Hence, we get * µ 1 + * µ 2 = d(SF b, π 3 ). erefore, condition (3) of eorem 1.2.1 is satis ed. It follows from eorem 1.2.1 and inequality (1.2.2) that for every Riemannian metric g conformal to SF b, π 3 , we have

L σ (g) v (g) ≤ 2b √ 3 + b -π 3 area(g)
with equality if and only if g is homothetic to SF b, π 3 .

We can also derive C. Bla er's similar result on the Mobius band M ( eorem 3 in [START_REF] Bla Er | Zur Riemannschen Geometrie im Grossen auf dem Möbiusband[END_REF]) using a minor adaptation of the previous proof.

Corollary 1.4.1. [Bl61, Satz 3] Let β > 0. en, for every Riemannian metric g on the Mobius band M of conformal type β, we have

sys(g) v (g) ≤ C β area(g)
where

C β =    e 2β +1 e 2β -1 arcsin e 2β -1 e 2β +1 if 0 < β ≤ ln 2 + √ 3 2 3 . 3β+2π-3 ln(2+ √ 3) 2 √ 3+β-ln(2+ √ 3) if β > ln 2 + √ 3 .
Moreover, the equality is a ained if and only if g is proportional to the spherical metric

S b restricted to E b = {(u, v) ∈ C | |v| ≤ b}, for b satisfying β = 2 ln(tan( π 4 + b 2 )
), in the rst case, and to the spherical-at metric SF b, π 3 restricted to

E b = {(u, v) ∈ C | |v| ≤ b}, for b = 1 4 β + π 3 -1 2 ln(2 + √
3), in the second case.

Proof of the geometric Inequality of type σ v h

In the following, we prove the geometric inequality of type σ v h on the Klein bo le. We will consider the curve families S 1 (ω 2 ), S 2 and S 3 (ω 2 ).

Proof of eorem 1.1.9. Let β > 0. ere exists a unique ω 2 ∈ (0, π 2 ) such that

β = 2 ln(tan( π 4 + ω 2 2 ))+ 2 cos(ω 2 ) tan(ω 2 ) -ω 2 + tan 2 (ω 2 ) -ω 2 tan(ω 2 ) + ω 2 2 . (1.5.1)
We equip K with the metric SF b,ω 2 , where

b = tan(ω 2 ) + tan 2 (ω 2 ) -ω 2 tan(ω 2 ) + ω 2 2 .
(1.5.2) en, we consider the families of curves S 1 (ω 2 ), S 2 and S 3 (ω 2 ), cf. Section 1.2.2. Each curve in S 1 (ω 2 ), (resp. S 2 , resp. S 3 (ω 2 )) has length equal to π, (resp. 4b, resp. 2π cos(ω 2 )) with respect to the metric SF b,ω 2 . Hence, condition (1) of eorem 1.2.1 is satis ed. Let

l : [-ω 2 , ω 2 ] -→ R a → tan(|a|) π . cos 2 (a)-cos 2 (ω 2 ) b-ω 2 2b-ω 2 (cos 2 (a)-cos 2 (ω 2 )) 1 2
We de ne on the family S 1 (ω 2 ) the measure

µ 1 = l(a) da ⊗ dθ where l(a) =      l(a) if a ∈ [-ω 2 , ω 2 ] l(a -2b) if a ∈ [2b -ω 2 , 2b] l(a + 2b) if a ∈ [-2b, -2b + ω 2 ]
On the family S 2 , we de ne the measure

µ 2 = cos(ω 2 ). b -ω 2 2b -ω 2 du.
Finally, on S 3 (ω 2 ), we de ne the measure

µ 3 = b 2b -ω 2 dv.
e masses m i of the measures

µ i are m 1 = 4 sin(ω 2 ) -4ω 2 (b-ω 2 ) cos(ω 2 ) 2b-ω 2 , m 2 = π cos(ω 2 )(b-ω 2 ) 2b-ω 2 and m 3 = 2b(b-ω 2 ) 2b-ω 2 . Now, since σ (SF b,ω 2 ) = π, v (SF b,ω 2 ) = 4b, and h (SF b,ω 2 ) = 2π cos(ω 2 ), condition (2) of eorem 1.2.1 is satis ed. Next, it can be easily veri ed that * µ 2 = cos(ω 2 ) f b,ω 2 (v) . b -ω 2 2b -ω 2 d(SF b,ω 2 )
where f b,ω 2 is the unique one-variable function invariant by the translation (u, v) → (u, v + 2b) which agrees with cosine on [-ω 2 , ω 2 ] and equal to the constant cos(ω

2 ) on [ω 2 , 2b -ω 2 ]. Moreover, * µ 3 = 0 if |v| ≤ ω 2 or 2b -ω 2 ≤ |v| ≤ 2b b 2b-ω 2 • d(SF b,ω 2 ) if ω 2 ≤ |v| ≤ 2b -ω 2
Next, by Lemma 1.2.5, we have

χ {|v|≤ω 2 } * µ 1 = 2χ {|v|≤ω 2 } ω 2 |v| cos 2 (v) -cos 2 (a) -1 2 l(a) da d(SF b,ω 2 )
en, we derive

χ {|v|≤ω 2 } * µ 1 = χ {|v|≤ω 2 } (1 - cos(ω 2 )(b -ω 2 ) f b,ω 2 (v)(2b -ω 2 ) ) d(SF b,ω 2 ).
Finally, since f b,ω 2 and l are invariant by the translation

(u, v) → (u, v + 2b), we have * µ 1 = (1 -cos(ω 2 )(b-ω 2 ) f b,ω 2 (v)(2b-ω 2 ) ) d(SF b,ω 2 ) if |v| ≤ ω 2 or 2b -ω 2 ≤ |v| ≤ 2b 0 if ω 2 ≤ |v| ≤ 2b -ω 2 us, we obtain * µ 1 + * µ 2 + * µ 3 = dSF b,ω 2 . Hence, condition (3) of eorem 1.2.1 is satis ed.
Hence, area(g)

3 2 ≥ 3 √ 3 √ π • (b -ω 2 ) (b -ω 2 )b (b 4 -4bω 2 + ω 2 2 + ω 2 4 -2b 2 (-2 + ω 2 2 )) 1 4 (2b -ω 2 ) σ (g) v (g) h (g)
with equality if and only if g is homothetic to SF b,ω 2 .

1.6 SOME REMARKS ON THE EXTREMAL METRICS

Some remarks on the extremal metrics

In this chapter, we present four families of conformally extremal metrics for di erent types of inequalities on the Klein bo le. ey are as follows.

1. e spherical metric S b , where β = 2 ln(tan( π 4 + b 2 )), which is conformally extremal for the inequality of type σ v for 0 < β ≤ 2 ln tan( π 4 + a 0 2 ) , and for the inequality of type

L σ v for 0 < β ≤ 2 ln(2 + √ 3).
2. e spherical-at metric SF b,ω 1 , where ω 1 and b verify the equations (1.3.1) and (1.3.2), which is conformally extremal for the inequality of type σ v for 2 ln tan( π 4 + a 0 2 ) < β < ∞.

3.

e spherical-at metric SF b, π 3 , where β = 2 ln(2

+ √ 3) + 4(b -π 3 ), which is conformally extremal for the inequality of type L σ v for 2 ln(2 + √ 3) < β < ∞.
4. e spherical-at metric SF b,ω 2 , where ω 2 and b verify the equations (1.5.1) and (1.5.2), which is conformally extremal for the inequality of type σ v h for any β.

Remark 1.6.1. We emphasize the fact that the metrics SF b,ω 1 and SF b,ω 2 do not have the same conformal type since ω 1 veri es the equation (1.3.1) whereas ω 2 veri es the equation (1.5.1). Note that the equality ω 1 = ω 2 leads to tan 2 (ω 1 ) -ω 1 tan(ω 1 ) + ω 1 2 + ω 1 = 0 which is not possible.

Moreover, the β-extremal metric (extremal metric in the conformal class of β) for the inequality of type σ v h agrees with the β-extremal metric for the inequality of type L σ v if and only if

β = β 0 = 2 ln(2 + √ 3) + 4( √ 3 + (3 - π √ 3 + π 2 9 ) 1 2 - π 3 ) (i.e., b = √ 3 + (3 -π √ 3 + π 2 9 )
1 2 and ω 2 = π 3 ). On the other hand, the β-extremal metric for the inequality of type σ v agrees with the β-extremal metric for the inequality of type L σ v if and only if β ≤ 2 ln(2 + √ 3). Finally, the β-extremal metric for the inequality of σ v and the β-extremal metric for the inequality of type σ v h never agree.

β (0, 2 ln(2 + √ 3)] (2 ln(2 + √ 3), 2 ln tan( π 4 + a 0 2 ) ] (2 ln tan( π 4 + a 0 2 ) , +∞) σ v ≤ C β area S b S b SF b,ω 1 L σ v ≤ C β area S b SF b, π 3 SF b, π 3 σ v h ≤ C β area 3 2 SF b,ω 2 SF b,ω 2 SF b,ω 2
Chapter 2

Optimal systolic inequalities on Finsler Mobius bands Abstract. We prove optimal systolic inequalities on Finsler Mobius bands relating the systole and the height of the Mobius band to its Holmes-ompson volume. We also establish an optimal systolic inequality for Finsler Klein bo les with symmetries, which we conjecture to hold true for arbitrary Finsler metrics. We describe extremal metric families both in both cases. where g runs over all the Riemannian metrics on M (hence the subscript R for Riemannian).

Introduction

us, σ R (T 2 ) = √ 3 2 . Following this direction, P. Pu [START_REF] Pu | Some inequalities in certain non-orientable Riemannian manifolds[END_REF] showed that σ R (RP 2 ) = 2 π , where the in mum is a ained exactly by the Riemannian metrics with constant (positive) curvature on the projective plane RP 2 . In the eighties, C. Bavard [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF]

proved that σ R (K 2 ) = 2 √ 2
π , where the in mum on the Klein bo le K 2 is not a ained by a smooth Riemannian metric. See also [START_REF] Sakai | A proof of the isosystolic inequality for the Klein bo le[END_REF], [START_REF] Bavard | Inégalités isosystoliques conformes pour la bouteille de Klein[END_REF] and [START_REF] Bavard | Une remarque sur la géométrie systolique de la bouteille de Klein[END_REF] for other proofs and variations on this inequality. ese are the only nonsimply connected closed surfaces with a known systolic area. e existence of extremal metrics in higher dimension is wide open. e original proofs of the optimal Riemannian systolic inequalities on T 2 , RP 2 and K 2 rely on the conformal representation theorem (a consequence of the uniformization theorem on Riemann surfaces) and proceed as follows. By the uniformization theorem, every Riemannian metric g on a closed surface is conformally equivalent to a Riemannian metric g 0 of constant curvature. Taking the average of g over the isometry group of g 0 gives rise to a new metric ḡ with the same area as g. By the Cauchy-Schwarz inequality, the systole of ḡ is at most the systole of g. us, the new metric ḡ has a lower ratio area/sys 2 than the original metric g. Now, if the isometry group of g 0 is transitive, which is the case for T 2 and RP 2 , the metric ḡ has constant curvature. Hence the result for the projective plane. en, it is not di cult to nd the extremal metric among at torus. e case of the Klein bo le requires an extra argument since the isometry group of g 0 is not transitive, cf. Section 2.7.

In this chapter, we consider Finsler systolic inequalities. Loosely speaking, a Finsler metric F is de ned as a Riemannian metric except that its restriction to a tangent plane is no longer a Euclidean norm but a Minkowski norm, cf. Section 2.2. From a dynamical point of view, the function F 2 can be considered as a Lagrangian which induces a Lagrangian ow on the tangent bundle T M of M . us, Finsler manifolds can be considered as degree 2 homogeneous Lagrangian systems. e trajectories of the Lagrangian correspond to the geodesics of the Finsler metric.

ere exist several de nitions of volume for Finsler manifolds which coincide in the Riemannian case. We will consider the Holmes-ompson volume vol HT , cf. Section 2.2. As previously, we can de ne the systolic area σ F , with the subscript F for Finsler, by taking the in mum in (2.1.2) over all Finsler metrics on M .

Contrary to the Riemannian case, there is no uniformization theorem for Finsler surfaces. As a result, the classical Riemannian tools to prove optimal systolic inequalities on surfaces, which are based on the conformal length method described above, do not carry over to the Finsler case.

New methods are thus required to deal with Finsler metrics.

e rst optimal Finsler systolic inequality has been obtained by S. Ivanov [START_REF] Ivanov | On two-dimensional minimal llings[END_REF][START_REF] Ivanov | Filling minimality of Finslerian 2-discs[END_REF] who extended Pu's systolic inequality to Finsler projective planes.

eorem 2.1.1 [START_REF] Ivanov | On two-dimensional minimal llings[END_REF][START_REF] Ivanov | Filling minimality of Finslerian 2-discs[END_REF]. Let RP 2 be a Finsler projective plane. en

vol HT (RP 2 ) sys 2 (RP 2 ) ≥ 2 π .
Furthermore, equality holds if all the geodesics are closed of the same length.

In particular, the systolic area of the projective plane is the same in the Riemannian and Finsler se ings, that is,

σ R (RP 2 ) = σ F (RP 2 ) = 2 π .
Note that eorem 2.1.1 provides an alternate proof of Pu's inequality in the Riemannian case which does not rely on the uniformization theorem.

Using a di erent method based on [START_REF] Gromov | Metric structures for Riemannian and non-Riemannian spaces[END_REF] Observe that σ F (T 2 ) = σ F (RP 2 ) contrary to the Riemannian case. An optimal Finsler systolic inequality holds for non-reversible Finsler metrics on T 2 , cf. [ABT]. Note also that there is no systolic inequality for non-reversible Finsler two-tori if one considers the Busemann volume instead of the Holmes-ompson volume, cf. [AB].

No systolic inequality holds for manifolds with boundary either. However, P. Pu [START_REF] Pu | Some inequalities in certain non-orientable Riemannian manifolds[END_REF] and C. Bla er [START_REF] Bla Er | Zur Riemannschen Geometrie im Grossen auf dem Möbiusband[END_REF] obtained optimal Riemannian systolic inequalities in each conformal class of the Mobius band and described the extremal metrics, cf. Section 2.4. Later, these inequalities were used by C. Bavard [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF] and T. Sakai [START_REF] Sakai | A proof of the isosystolic inequality for the Klein bo le[END_REF] in their proofs of the systolic inequality on the Klein bo le in the Riemannian case. e proof of the optimal conformal Riemannian systolic inequalities on the Mobius band relies on the uniformization theorem and the conformal length method (as in the original proofs of the Riemannian systolic inequalities on T 2 , RP 2 and K 2 ).

In this chapter, we rst prove a Finsler generalization of the optimal systolic inequality on T 2 extending Loewner's inequality, cf. [START_REF] Keen | An extremal length on a torus[END_REF], and derive further optimal geometric inequalities on Finsler cylinders, cf. Section 2.3. ese results allow us to establish an optimal inequality on every Finsler Mobius band M relating its systole sys(M), its height h(M) and its (Holmesompson) volume vol HT (M) at least when M is wide enough, cf. Section 2.5. Here, the height h(M) represents the minimal length of arcs with endpoints on the boundary ∂M, which are not homotopic to an arc in ∂M, cf. De nition 2.3.4. More precisely, we prove the following.

eorem 2.1.3. Let M be a Finsler Mobius band. Let λ := h(M) sys(M) . en

vol HT (M) sys(M) h(M) ≥    2 π if λ ∈ (0, 1] 1 π λ+1 λ
otherwise.

(2.1.3) Moreover, the above inequalities are optimal for every value of λ ∈ (0, +∞).

We describe extremal and almost extremal metric families in details in Section 2.4, Example 2.5.4

and Example 2.6.2.

e optimal Finsler systolic inequality on the Klein bo le is still unknown. However, based on the inequality (2.1.3) on Finsler Mobius bands, we obtain a partial result for Finsler Klein bo les with nontrivial symmetries. We refer to De nition 2.7.1 for a description of the symmetries considered in the statement of the following theorem.

eorem 2.1.4. Let K be a Finsler Klein bo le with a soul, soul-switching or rotational symmetry.

en vol HT (K) sys 2 (K) ≥ 2 π . (2.1.4)
Moreover, the inequality is optimal.

We also present some extremal metric family in Example 2.7.7.

Finally, we present as a conjecture that the inequality (2.1.4) should hold for every Finsler Klein bo le with or without symmetries. at is, σ F (K) should be equal to 2 π (as σ F (T 2 ) and σ F (RP 2 )). Note, however, that the Finsler systolic area σ F is not the same for all the surfaces: it goes to in nity with the genus of the surface, cf. [START_REF] Gromov | Filling Riemannian manifolds[END_REF].

Preliminaries

In this section, we introduce general de nitions regarding Finsler manifolds.

A (reversible) Finsler metric F : T M → [0, +∞) on the tangent bundle T M of a smooth ndimensional manifold M is a continuous function satisfying the following conditions (for simplicity, let F x := F | TxM ):

1. Smoothness: F is smooth outside the zero section; 2. Homogeneity: F x (tv) = |t|F x (v) for every v ∈ T x M and t ∈ R;

3.

adratic convexity: for every x ∈ M , the function F 2 x has positive de nite second derivatives on T x M \ 0, i.e., if p, u, v ∈ T x M , the symmetric bilinear form

g p (u, v) := 1 2 ∂ 2 ∂s∂t F 2 x (p + tu + sv) | t=s=0
is an inner product.

e pair (M, F ) is called a Finsler manifold. If F is only positive homogeneous instead of homogeneous, that is, (2) only holds for t ≥ 0, we say that the Finsler metric is non-reversible. For simplicity, we will only consider reversible Finsler metrics.

Conditions (1), (2) and (3) imply that F is strictly positive outside the zero section and that for every x ∈ M and u, v ∈ T x M , we have

F x (u + v) ≤ F x (u) + F x (v),
with equality if and only if u = λv or v = λu for some λ ≥ 0, cf.

• x = {u ∈ T x M | u, v ≤ 1 for every v ∈ B x } of B x ,
where ., . is a given scalar product on T x M .

In the Riemannian case, there exists a unique notion of volume, up to normalization, which agrees both with the n-dimensional Hausdor measure determined by the Riemannian metric and with the projection of the Liouville measure from the unit tangent bundle, cf. [BBI01, §5.5]. However, in the Finsler case, there is no notion of volume that satis es both properties, cf. [START_REF] Burago | Minimality of planes in normed spaces[END_REF]. is leads to two distinct notions of Finsler volume presented below.

Denote by ε n the Euclidean volume of the Euclidean unit ball in R n . Let dx represent a given volume form on M and m be the restriction of this volume form to each tangent space T x M .

Similarly, let m * be the restriction of the volume form dual to dx to each cotangent space T *

x M . e Busemann volume, cf. [START_REF] Busemann | Intrinsic area[END_REF], is de ned as

vol B (M ) := M ε n m(B x ) dx. ( 2

A systolic inequality on Finsler two-tori

In this section we establish a Finsler version of the Minkowski second theorem for the two-torus.

More precisely, L. Since there is no uniformization theorem for Finsler metrics, the proof of this proposition di ers from the proof of Proposition 2.3.1.

Proof. Let α be a systolic loop of T 2 and β be the shortest closed curve of T 2 homologically independent with α. Denote by a and b the lengths of α and β. e loops α and β are simple and intersect each other at a single point. Cu ing T 2 open along α and β gives rise to a surface ∆ isometric to a fundamental domain of T 2 . Let L be a positive number greater than max{a, b}.

Denote by p and q the smallest integers such that pa ≥ L and qb ≥ L. en, glue pq copies of ∆ in such a way that the resulting shape is isometric to the fundamental domain of a Finsler torus of volume pq times the volume of T 2 and of systole equal to min{pa, qb}. By construction, this new Finsler torus is a degree pq cover of T 2 . en, by eorem 2.1.2, we have

pq vol HT (T 2 ) ≥ 2 π (min{pa, qb}) 2 .
Hence,

vol HT (T 2 ) ≥ 2 π L p L q ≥ 2 π p -1 p q -1 q ab.
By choosing L large enough, the integers p and q can be made arbitrarily large, which leads to the desired inequality. Now, if T 2 is the quotient of R 2 , endowed with a parallelogram norm, by a la ice generated by two vectors of lengths a and b which are parallel to the sides of the unit ball of the parallelogram norm, then vol HT (T 2 ) = 2 π ab.

Remark 2.3.3. Brie y speaking, the idea of the proof of Proposition 2.3.2 is to use nite covers to get a quasi-isosystolic two-torus (i.e., whose rst homology group is generated by two loops of lengths nearly the systole) and to apply the systolic inequality of eorem 2.1.2 to this two-torus.

is argument also applies in the Riemannian case and gives an alternative proof of Proposition 2.3.1 without the use of the uniformization theorem.

We can apply Proposition 2.3.2 to prove a systolic inequality on Finsler cylinders. First, we give the following de nition De nition 2.3.4. Let M be a compact Finsler surface with boundary. e height h(M ) of M is the minimal length of arcs with endpoints on the boundary ∂M , which are not homotopic to an arc in ∂M . More formally,

h(M ) := inf{ (γ)|γ : [0, 1] → M with γ(0), γ(1) ∈ ∂M and [γ] = 0 ∈ π 1 (M, ∂M )}.
A height arc of M is a length-minimizing arc of γ of M with endpoints in ∂M inducing a nontrivial class in π 1 (M, ∂M ). By de nition, the length of a height arc of M is equal to h(M ).

Proposition 2.3.5. Let C be a Finsler cylinder. en,

vol(C) ≥ 2 π sys(C)h(C).
Proof. Let k be a positive even integer. We glue k copies of C by identi ying the identical boundary components pairwise. e resulting space is a torus T 2 . Every loop of T 2 non freely homotopic to a multiple of a boundary component of C is of length at least k h(C). erefore, for symmetry reasons, if k satis es k h(C) ≥ sys(C), the systole of the torus T 2 is equal to the systole of the cylinder C. Applying Proposition 2.3.2 to this torus, we derive

vol(T 2 ) = k vol(C) ≥ 2 π k sys(C)h(C).
Hence the result.

We will make use of Proposition 2.3.5 in the proof of eorem 2.1.3 for wide Finsler Mobius bands, cf. Section 2.5.

Natural candidates for extremal metrics

In this section, we rst review the extremal Riemannian metrics for systolic inequalities on the Mobius band and the Klein bo le presented in [Pu52, Bl61, Ba86, Ba88, Sak88]. By analogy with the Riemannian metrics, we construct Finsler metrics which are natural to consider when studying optimal Finsler systolic inequalities.

Consider the standard sphere S 2 . Denote by u and v the longitude and the latitude on S 2 . Let a ∈ (0, π 2 ). e a-tubular neighborhood of the equator {v = 0} is a spherical band S a which can be represented as

S a := {(u, v) | -π ≤ u ≤ π, -a ≤ v ≤ a}.
e quotient of S a by the antipodal map is a Riemannian Mobius band with curvature 1 denoted by M a . e conformal modulus space of the Mobius band is parametrized by M a with a ∈ (0, π 2 ). More precisely, every conformal class on the Mobius band agrees with the conformal structure induced by some M a with a ∈ (0, π 2 ). Furthermore, the conformal classes of the M a 's are pairwise distinct. Indeed, the (conformal) stereographic projection sends S a to a planar annulus whose conformal modulus varies from 0 to ∞ as a runs over (0, π 2 ), cf. In [START_REF] Bla Er | Zur Riemannschen Geometrie im Grossen auf dem Möbiusband[END_REF], C. Bla er obtained optimal lower bounds for the functionals

σ 1 := vol sys 2 - , σ 2 := vol sys -× v
and σ 3 := vol sys × v in each conformal class of the Mobius band. More precisely, for every Riemannian metric conformally equivalent to M a , we have the sharp lower bound

σ 1 (M) ≥ σ 1 (M a ).
(2.4.1)

We also have the sharp inequality

σ 2 (M) ≥    σ 2 (M a ) if a ∈ (0, b] σ 2 (M α(a) ∪ C a,α(a) ) if a ∈ [b, π 2 ) (2.4.2)
where b is the unique solution in (0, Finally, we have the third sharp inequality

σ 3 (M) ≥    σ 3 (M a ) if a ∈ (0, π 3 ] σ 3 (M π 3 ∪ C a, π 3 ) if a ∈ [ π 3 , π 2 ) (2.4.3)
With the help of (2.4.1), C. Bavard [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF] established the optimal isosystolic inequality on the Klein bo le. Later, T. Sakai [START_REF] Sakai | A proof of the isosystolic inequality for the Klein bo le[END_REF] used the inequalities (2.4.2) and (2. With this observation in mind, it is natural to consider the following (non-quadratically convex)

Finsler metrics as potential extremal metrics. e idea is to adjust the shapes of the unit balls in the tangent bundle of the Mobius band so that the systolic and meridian directions ll in the unit tangent bundle. More precisely, de ne a Finsler metric F a on S a whose restriction to each tangent plane T x S a is a norm F a | x of unit ball B x given by the convex hull of the systolic directions of M a , cf. Figure 2.2. In longitude and latitude coordinates, the ball B x at x = (u, v) can be represented as Hence, the Finsler metric F a can be represented in local coordinates as

B x := {(ξ u , ξ v ) ∈ T x S a | ξ 2 u + ξ 2 v ≤ 1, |ξ v | ≤ sin θ(v)}.
F a =    1 sin(θ(v)) |dv| if arctan dv du ∈ [θ(v), π 2 ] √ du 2 + dv 2 if arctan dv du ∈ [0, θ(v)]
is metric passes to the quotient by the antipodal map to a Finsler metric still denoted by F a .

Denote by M Fa the Finsler Mobius band so obtained.

Instead of considering v , whose de nition relies on conformal data, we will work with the height h(M), cf. De nition 2.3.4, in the Finsler case.

Some geometric features of the Finsler Mobius bands M Fa are summarized in the following two propositions.

Proposition 2.4.1. Let a ∈ (0, π 2 ). en, sys -(M Fa ) = π, sys + (M Fa ) = 2π cos(a) and h(M Fa ) = π.

In particular, if a = π 3 then sys(M Fa ) = sys

+ (M Fa ) = sys -(M Fa ) = h(M Fa ) = π.
Proof. Let us start with a useful observation. Denote by S the interior of the domain of U M Fa formed by the unit tangent vectors of the great circles of M a . e Finsler metric F a coincides with the round Riemannian metric of M a on S. erefore, the subset S is stable under the geodesic ow of F a (which is well-de ned on S). Furthermore, the length of a great circle with respect to F a is equal to π.

Let us show that h(M Fa ) = π. Consider a height arc γ of M Fa . e arc γ can be parametrized with respect to the latitude. Otherwise, we could remove a subarc of γ joining two points at the same latitude and still make up an arc in the same relative homotopy class as γ with the remaining pieces using the rotational symmetry of M Fa . is would contradict the length-minimizing property of γ. Hence,

h(M Fa ) = (γ) = a -a 1 sin θ(v) dv = a -a cos(v) cos 2 (v) -cos 2 (a) dv = 2 arctan √ 2 sin(v) cos(2v) -cos(2a) a 0 = π.
Now, let us show that the systolic curves of M Fa agree with the great circles of M a in the nonorientable case and with the boundary of M a in the orientable case. Consider an orientable or nonorientable noncontractible loop γ of minimal length in M Fa .

If γ lies in the boundary of M Fa then the loop γ is orientable of length 2π cos(a). us, we can assume that γ passes through an interior point p of M Fa .

If a tangent vector of γ lies in S then the geodesic arc γ coincides with a great circle of M a in the nonorientable case and with a great circle run twice in the orientable case. (Recall that S is stable by the geodesic ow of F a .) In the former case, the curve γ is of length π, while in the la er, it is of length 2π. us, we can assume that the tangent vectors of γ do not lie in S.

Consider the closed li γ of γ in S a . Let c ± be the two extreme great circles of S a passing through the li s of p and tangent to the boundary of S a . at is, c ± are the great circles of S a making an angle of ±θ(v) with the curves of constant latitude ±v in S a passing through the li s of p.

Since the tangent vectors of γ do not lie in S, the curve γ does not intersect c ± in the interior of S a , except at the li s of p. erefore, there exists a subarc of γ (actually two subarcs of γ)

joining the two boundary components of S a in the region delimited by the great circles c ± and the boundary of S a , see the gray region of Figure 2.3. us, (γ) ≥ h(M Fa ) = π with equality if γ agrees with c ± .

We conclude that sys -(M Fa ) = π and sys + (M Fa ) = (∂M Fa ) = 2π cos(a). Hence,

sys(M Fa ) =    sys -(M Fa ) if a ∈ (0, π 3 ] sys + (M Fa ) if a ∈ [ π 3 , π 2 )
Remark 2.4.2. e Finsler Mobius bands M Fa are not pairwise isometric since they have distinct orientable systoles.

Proposition 2.4.3. Let a ∈ (0, π 2 ). en, vol HT (M Fa ) = 2π. (2.2.2), we have

vol HT (M Fa ) = 1 2π Sa m(B * x ) dm(x) = 2 π π 2 -π 2 a -a θ(v) + 1 tan θ(v) cos(v) du dv = 2 π π 2 -π 2 a -a arccos cos(a) cos(v) + cos(a) cos 2 (v) -cos 2 (a) cos(v) du dv = 2π.
Remark 2.4.4. As a consequence of Propositions 2.4.1 and 2.4.3, we observe the following couple of points.

1. e orientable and nonorientable systoles of F a agree with those of its dual F * a . Hence, sys(M Fa ) = sys(M F * a ). Moreover, computations similar to those in Propositions 2.4.1 and 2.4.3 show that h(M F * a ) = π(1 -cos(a)) and vol HT (M F * a ) = 2π sin 2 (a). is means that for both F a and its dual F * a , we have vol HT (M) sys -(M) h(M) → 2 π , when a → π 2 .

2. e Finsler Mobius bands M Fa with a ∈ (0, π 3 ] a ain the equality case in (2.1.3) when sys(M) = h(M).

Systolic inequalities on wide Finsler Mobius bands

In this section, we give a proof of eorem 2.1.3 for wide Finsler Mobius bands, that is, when λ ≥ 1. More precisely, we prove the following result.

Proposition 2.5.1. Let M be a Finsler Mobius band with h(M) ≥ sys(M). en

vol HT (M) ≥ 1 π sys(M) (sys(M) + h(M)).
(2.5.1)

We present examples showing this result is optimal at the end of this section, cf. Example 2.5.4.

Proof.

Consider

U := {x ∈ M | d(x, ∂M) ≤ λ -1 2 sys(M)}.
Slightly perturbing the distance function d(., ∂M) if necessary, we can assume that this distance function is a Morse function on M for which λ-1 2 sys(M) is a regular value. In this case, U is a surface with boundary. If M has some "big bumps", the surface U may possibly have some holes. More precisely, the surface U may not be a topological cylinder as some of its boundary components may bound topological disks in M.

Let U be the union of U with the topological disks of M bounded by the boundary components of U. Under this construction, U is a cylinder one of whose boundary components agrees with ∂M. Clearly, the height of U is equal to λ-1 2 sys(M). Furthermore, since the inclusion U ⊂ M induces a π 1 -isomorphism, we have sys( U) ≥ sys(M). Applying Proposition 2.3.5 to the cylinder U

yields vol HT ( U) ≥ λ -1 π sys(M) 2 . (2.5.2)
Now, consider the Finsler Mobius band M -:= M \ U.

Lemma 2.5.2. e height and systole of M -satisfy h(M -) = sys(M) and sys(M -) ≥ sys(M).

Proof. Let γ -be a height arc of M -, cf. De nition 2.3.4. By construction, ∂ U = ∂M -∪ ∂M and the points of ∂M -are at distance λ-1 2 sys(M) from ∂M. erefore, the two endpoints of γ - can be connected to ∂M by two arcs γ 1 and γ 2 of U, each of length λ-1 2 sys(M). Moreover, the arc γ := γ -∪ γ 1 ∪ γ 2 with endpoints in ∂M induces a nontrivial class in π 1 (M, ∂M). erefore, since h(M) = λ sys(M), we obtain

h(M -) = (γ) -(λ -1) sys(M) ≥ h(M) -(λ -1) sys(M) ≥ sys(M).
Now, let γ be a height arc of M. By de nition, we have (γ) = h(M) = λ sys(M). e part γ ∩ U of γ in U is made of two arcs, each of length at least λ-1 2 sys(M). Moreover, the arc γ ∩ M -with endpoints in ∂M -induces a nontrivial class in π 1 (M -, ∂M -). Hence,

h(M -) ≤ (γ) -(γ ∩ U) ≤ h(M) -(λ -1) sys(M) ≤ sys(M).
Since the inclusion M -⊂ M induces a π 1 -isomorphism, we obtain sys(M -) ≥ sys(M).

Consider the projective plane RP 2 de ned as the quotient M -/∂M -, where the boundary ∂M - is collapsed to a point. Strictly speaking, the Finsler metric on RP 2 has a singularity at the point to which ∂M -collapses, but we can smooth it out.

e following result allows us to derive the systole of RP 2 . Lemma 2.5.3. Let RP 2 be the projective plane de ned as the quotient M/∂M of a Finsler Mobius band M. en, sys(RP 2 ) = min {h(M), sys(M)}

where RP 2 is endowed with the quotient metric.

Proof. Let γ be a noncontractible loop in RP 2 . e curve γ li s either to a noncontractible loop in M or to a noncontractible arc in M joining two points of the boundary ∂M. In the former case, the length of γ is at least sys(M), while in the la er, it is at least h(M). On the other hand, we can easy construct noncontractible loops in RP 2 of length sys(M) or h(M).

From Lemma 2.5.3 and Lemma 2.5.2, the systole of RP 2 is equal to sys(M). Applying eorem 2.1.1 to RP 2 , we obtain

vol HT (M -) = vol HT (RP 2 ) ≥ 2 π sys 2 (M).
is inequality combined with (2.5.2) yields

vol HT (M) = vol HT ( U) + vol HT (M -) ≥ 1 + λ π sys(M) 2 .
Hence the result.

We conclude this section by describing extremal and almost extremal Finsler metrics when λ ≥ 1.

Systolic inequalities on narrow Finsler Mobius bands

In this section, we give a proof of eorem 2.1.3 for narrow Finsler Mobius bands, that is, when λ < 1. More precisely, we prove the following result.

Proposition 2.6.1. Let M be a Finsler Mobius band with h(M) < sys(M). en

vol HT (M) ≥ 2 π sys(M) h(M).
is inequality is optimal. Extremal Finsler metrics can be constructed as follows.

Example 2.6.2. Let λ < 1. Endow the plane R 2 with the sup-norm. e quotient of the strip

R × [-π 2 λ, π 2
λ] by the group generated by the map (x, y) → (x + π, -y) is a Finsler Mobius band M with vol HT (M) = 2λπ, sys(M) = π and h(M) = λπ.

Before proceeding to the proof of this proposition, we need to introduce a few de nitions and notions.

De nition 2.6.3. e orientable double cover of a Finsler Mobius band M is a cylinder denoted by C. e points of C which are at the same distance from each boundary component of C form a simple closed curve c invariant under deck transformations. e soul of M, denoted by σ, is de ned as the projection of c to M. Note that σ is a nonorientable simple closed curve of M whose homotopy class generates π 1 (M).

Let K be the Finsler Klein bo le obtained by a aching to M another copy M of M along their boundary. ( e Finsler metric on K may have a singularity line along ∂M.) e isometry of K switching the souls of M and M , and leaving ∂M pointwise xed is called the soul-switching symmetry of K.

Let s, h ∈ R + . e systole-height inequality on the Mobius band is said to be satis ed for (s, h) if for every Finsler Mobius band M with sys(M) ≥ s and h(M) ≥ h, we have

vol HT (M) ≥ 2 π sh.
By scale invariance, if the systole-height inequality is satis ed for (s, h), then it is also satis ed for (s , h ) with h s = h s .

We rst prove the following preliminary result.

Lemma 2.6.4. Let λ ∈ (0, 1]. Suppose that the systole-height inequality on the Mobius band is satis ed for (s, h) with h s = λ. en, it is also satis ed for (s, h) with h s = λ 2 .

Proof. Let M be a Finsler Mobius band with sys(M) ≥ s and h(M) ≥ h, where s h = λ 2 . Consider the Klein bo le K made of two copies of M de ned in De nition 2.6.3 and cut it open along the soul σ of M . e resulting surface is a Finsler Mobius band denoted by 2M whose boundary component double covers σ in K.

Let α be a noncontractible loop of 2M. Decompose α into two parts a = α ∩ M and a = α ∩ M with α = a ∪ a . e parts a and a form two collections of arcs with endpoints lying in ∂M = ∂M . By construction, the image a of a by the soul-switching symmetry lies in M.

Furthermore, the union ᾱ = a ∪ a forms a closed curve lying in M and homotopic to α in 2M (and so noncontractible in M). Since ᾱ has the same length as α, we conclude that sys(2M) ≥ sys(M) ≥ s.

Actually, since the inclusion M ⊂ 2M is a strong deformation retract, we derive the relation sys(2M) = sys(M). But we will not make use of this equality in the sequel.

By construction, the distance between the soul σ and ∂M (and between σ and ∂M ) is at least 1 2 h(M). is implies that h(2M) ≥ 2h(M) ≥ 2h.

Actually, we can show that h(2M) = 2h(M) (but we will not make use of this relation a erwards). Indeed, let α be a height arc of M. By de nition, (α) = h(M). Denote by α its image in M by the soul-switching symmetry of K. e trace of the union α ∪ α to 2M de nes an arc with endpoints in ∂(2M) inducing a nontrivial class in π 1 (2M, ∂(2M)). e length of this arc is twice the length of α. erefore, h(2M) ≤ 2h(M). Hence, the equality h(2M) = 2h(M).

In conclusion, the Mobius band 2M satis es sys(2M) ≥ s and h(2M) ≥ 2h. Since 2h s = λ, the systole-height inequality is satis ed for (s, 2h) by the lemma assumption. erefore,

2 vol HT (M) = vol HT (2M) ≥ 4 π sh
and the result follows.

We establish a second preliminary result.

Lemma 2.6.5. Let λ 1 , λ 2 ∈ R such that 0 < λ 1 < λ 2 ≤ 1. Suppose that the systole-height inequality on the Mobius band is satis ed for (s, h) with h s = λ 1 or λ 2 . en, it is also satis ed for (s, h) with h s = λ 1 +λ 2 2 .

Proof. Let M be a Finsler Mobius band with sys(M) ≥ s and h(M) ≥ h, where h s = λ 1 +λ 2 2 . Consider the Klein bo le K made of two isometric Mobius bands M and M with souls σ and σ de ned in De nition 2.6.3. Consider also

M 1 = {x ∈ K | λ 2 d(x, σ) ≤ λ 1 d(x, σ )} and M 2 = {x ∈ K | λ 2 d(x, σ) ≥ λ 1 d(x, σ )}
Note that if we drop the multiplicative constants λ 1 and λ 2 in the de nitions of M 1 and M 2 , we obtain M and M . e subset M 1 is a Finsler Mobius band contained in M. Similarly, the subset M 2 is a Finsler Mobius band containing M . Observe also that the Mobius bands M 1 and M 2 cover K and that their interiors are disjoint. Every point z ∈ ∂M i satis es the equality

λ 2 d(z, σ 1 ) = λ 1 d(z, σ 2 ).
(2.6.1) By symmetry, the distance between σ 1 = σ and σ 2 = σ is equal to h(M). It follows by the triangle inequality that

d(z, σ 1 ) + d(z, σ 2 ) ≥ h(M). (2.6.2)
As a result of the relations (2.6.1) and (2.6.2), we obtain

d(z, σ i ) ≥ λ i λ 1 + λ 2 h(M).
(2.6.3) Now, let α be an arc of M i with endpoints x, y ∈ ∂M i inducing a nontrivial class in π 1 (M i , ∂M i ).

As α intersects σ i , we deduce from (2.6.3) that

(α) ≥ d(x, σ i ) + d(y, σ i ) ≥ 2λ i λ 1 + λ 2 h(M). erefore, h(M i ) ≥ 2λ i λ 1 + λ 2 h(M) ≥ 2λ i λ 1 + λ 2 h.
(2.6.4)

In another direction, we can also bound from below the systole of M 1 and M 2 as follows.

For the systole of M 1 , since the inclusion M 1 ⊂ M induces a π 1 -isomorphism, we derive sys(M 1 ) ≥ sys(M) ≥ s.

(2.6.5)

Note that the rst inequality may be strict as the inclusion M 1 ⊂ M is strict.

For the systole of M 2 , we argue as in Lemma 2.6.4. Let α be a noncontractible loop of M 2 .

Decompose α into two parts a = α ∩ M and a = α ∩ M with α = a ∪ a . e union ᾱ = a * ∪ a , where a * is the image of a by the soul-switching symmetry of K, forms a closed curve of length (α) lying in M and homotopic to α in M 2 . Hence, sys(M 2 ) ≥ sys(M) ≥ s.

(2.6.6) e systole-height inequality on the Mobius band is satis ed for (s, 2λ i λ 1 +λ 2 h) from the lemma assumption since 2λ i λ 1 +λ 2 h s = λ i . From the bounds (2.6.4), (2.6.5) and (2.6.6), this inequality applies to M i and yields

vol HT (M i ) ≥ 4 π λ i λ 1 + λ 2 sh.
(2.6.7)

Finally, recall that the Mobius bands M 1 and M 2 cover K and that their interiors are disjoint. By adding up (2.6.7) for i = 1, 2, we conclude that

2 vol HT (M) = vol HT (K) = vol HT (M 1 ) + vol HT (M 2 ) ≥ 4 π sh
Hence the result.

Remark 2.6.6. At rst glance, it seems more natural to assume that sys(M) = s and h(M) = h in the de nition of the systole-height inequality, cf. De nition 2.6.3. Observe that the proof of Lemma 2.6.4 carries over with this alternative notion. However, we have not been able to directly prove a result similar to Lemma 2.6.5 with this more restrictive notion. e reason is that the inequality (2.6.5), namely sys(M 1 ) ≥ sys(M), may be strict as the inclusion M 1 ⊂ M is strict. To get around this subtle di culty, we relaxed the original de nition and formulated the systole-height inequality in terms of lower bounds for the systole and the height of the Mobius band.

We can now proceed to the proof of Proposition 2.6.1.

Proof of Proposition 2.6.1. By Lemma 2.6.4, for every nonnegative integer k, the systole-height inequality on the Mobius band is satis ed for (s, h) with h s = 1 2 k . Combined with Lemma 2.6.5, this implies that the systole-height inequality is satis ed for every (s, h) where h s is a dyadic rational of (0, 1). Since the height, the systole and the volume are continuous over Finsler metrics, the result follows from the density of the dyadic rationals in [0, 1].

Systolic inequality on Finsler Klein bottles

In this section, we show that the systolic area of Finsler Klein bo les with soul, soul-switching or rotational symmetries is equal to 2 π .

De nition 2.7.1. Recall that every Riemannian Klein bo le is conformally equivalent to the quotient of R 2 by the isometry group G generated by the glide re ection (x, y) → (x + π, -y) A Finsler metric on K has a soul symmetry if its li to R 2 is invariant by the map (x, y) → (x, -y). Similarly, a Finsler metric on K has a soul-switching symmetry if its li to R 2 is invariant by the map (x, y) → (x, b -y).

Finally, a Finsler metric on K has a rotational symmetry if its li to R 2 is invariant by the map (x, y) → (x + θ, y) for every θ ∈ [0, 2π]. ese de nitions are consistent with the notions introduced in 2.6.

In 1986, C. Bavard established an optimal isosystolic inequality for Riemannian Klein bo les, cf. [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF]. Alternative proofs can be found in [START_REF] Sakai | A proof of the isosystolic inequality for the Klein bo le[END_REF][START_REF] Bavard | Inégalités isosystoliques conformes pour la bouteille de Klein[END_REF][START_REF] Bavard | Une remarque sur la géométrie systolique de la bouteille de Klein[END_REF]. All the proofs are based on the uniformization theorem. In fact, as mentioned in the introduction, the problem boils down to consider Riemannian Klein bo les invariant under soul (and rotational) symmetry. ese Klein bo les are made of two isometric copies of Riemannian Mobius bands. us, in the end, the systolic inequality on Riemannian Klein bo les follows from optimal systolic inequalities on Riemannian Mobius bands, cf. [START_REF] Pu | Some inequalities in certain non-orientable Riemannian manifolds[END_REF][START_REF] Bla Er | Zur Riemannschen Geometrie im Grossen auf dem Möbiusband[END_REF].

ere is no known optimal isosystolic inequality on the Klein bo le for Finsler metrics. However, we obtain the following partial result similar to the Riemannian case. Note that in the Riemannian case, the hypothesis is automatically satis ed by an average argument.

eorem 2.7.2. Let K be a Finsler Klein bo le with a soul, soul-switching or rotational symmetry.

en vol HT (K) sys 2 (K) ≥ 2 π .
(2.7.1)

Moreover, the inequality is optimal. Proof. e inclusion M ⊂ K induces a π 1 -isomorphism. Hence, sys(K) ≤ sys(M). Now, we only have two cases to consider.

Let K = R 2 /G
First, if h(M) ≤ sys(M), then we deduce from eorem 2.1.3, rst case, that

vol HT (K) sys 2 (K) ≥ vol HT (M) sys(M) h(M) ≥ 2 π .
Second, if h(M) ≥ sys(M), then we deduce from eorem 2.1.3, second case, that

vol HT (K) sys 2 (K) ≥ 1 π 1 + sys(M) h(M) sys(M) h(M) sys 2 (K) ≥ 1 π h(M) sys(K) + sys(M) sys(K) sys(M) sys(K) ≥ 2 π ,
since both h(M) sys(K) and sys(M) sys(K) are greater or equal to 1.

e next three lemmas show that the assumption of Lemma 2.7.3 is satis ed when the Finsler metric on K has soul, soul-switching or rotational symmetries.

Lemma 2.7.4. If K is a Finsler Klein bo le with a soul symmetry then h(M) ≥ sys(K).

Proof. Observe that the soul symmetry of K leaves both M and ∂M invariant. Given an arc α of M, we denote by α * the arc of M symmetric to α by the soul symmetry. Let γ be a height arc of M, cf. De nition 2.3.4. is arc decomposes into two subarcs α and β connecting ∂M to the soul of M with (α) ≤ (β). e arc α ∪ α * with endpoints in ∂M induces a nontrivial class in π 1 (M, ∂M) of length at most the length of γ = α ∪ β. By de nition of h(M), we conclude that α ∪ α * is as long as γ and so is length-minimizing in its relative homotopy class. In

Conjecture. Let K be a Finsler Klein bo le. en

vol HT (K) sys 2 (K) ≥ 2 π .
Moreover, the inequality is optimal.

Remark 2.7.8. If the conjecture is true, the Finsler systolic areas of RP 2 , T 2 and K would be the same.

A non-optimal systolic inequality on Finsler Klein bottles

In this section, we present a non-optimal systolic inequality on Finsler Klein bo les. Abstract. It is known that the genus two surface admits a piecewise at metric with conical singularities which is extremal for the systolic inequality among all nonpositively curved metrics.

We show that this piecewise at metric is also critical for slow metric variations, without curvature restrictions, for another type of systolic inequality involving the lengths of the shortest noncontractible loops in di erent free homotopy classes. e free homotopy classes considered correspond to those of the systolic loops and the second-systolic loops of the extremal surface.

Introduction

We are interested in optimal geometric inequalities relating the area of a closed Riemannian surface Σ to the lengths of the shortest loops in certain homotopy classes. A typical example is given by the systolic inequalities, which relate the area of the surface to the systole, that is, the length of the shortest noncontractible loop in Σ.

e rst known systolic inequality is due to C. Loewner in 1949, who proved that every Riemannian two-torus (T 2 , g) satis es

area(g) ≥ √ 3 2 sys 2 (g),
where sys(g) denotes the systole of the torus. Furthermore, the equality holds if and only if the torus is endowed with a at hexagonal metric. C. Loewner did not publish his result, however it was mentioned by his student P. Pu, cf. [START_REF] Pu | Some inequalities in certain non-orientable Riemannian manifolds[END_REF], who established a sharp systolic inequality on the projective plane RP 2 . In this case, the equality is a ained precisely by the round metrics.

More than thirty years later, C. Bavard, cf. [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF] (see also [START_REF] Sakai | A proof of the isosystolic inequality for the Klein bo le[END_REF]) proved a sharp systolic inequality on the Klein bo le K. Here, the extremal metrics are not smooth. ese are the only manifolds for which an optimal systolic inequality is known. All these systolic inequalities can be wri en as follows. If

sys(g) ≥ sys(g 0 ) then area(g) ≥ area(g 0 ) (3.1.1)
where g 0 is the extremal metric in each of these cases.

Optimal systolic-like inequalities, i.e., inequalities that relate the area to the product of the lengths of the shortest loops or arcs in di erent relative homotopy classes are known only for the twotorus T 2 , cf. We can reformulate a systolic-like inequality as in (3.1.1). For instance, let 1 (g) represent the length of the shortest noncontractible loop on (Σ, g), that is, Denote by Λ 1 the subset of the free homotopy classes generated by the systolic loops of (Σ, g)

and their multiples:

Λ 1 = { γ k | γ = 0, g 0 (γ) = 1 (g 0 ) and k ∈ Z * }. (3.1.2)
Similarly, let 2 (g) be the length of the shortest noncontractible loop on (Σ, g) which is not homotopic to a systolic loop or its multiples. at is,

2 (g) = inf γ / ∈Λ 1 ∪{0} g (γ).
Denote also by

Λ 2 = { γ | γ / ∈ Λ 1 and g 0 (γ) = 2 (g 0 )} (3.1.3)
the subset of the free homotopy classes generated by the shortest noncontractible loops of (Σ, g) which are not freely homotopic to systolic loops or their multiples. As a particular case of L. Keen's optimal systolic-like inequality for the two-torus in [START_REF] Keen | An extremal length on a torus[END_REF], we have the following.

If

1 (g) ≥ 1 (g 0 ) 2 (g) ≥ 2 (g 0 )
then area(g) ≥ area(g 0 ).

Recently, M. Katz and S. Sabourau, cf. [START_REF] Katz | An optimal systolic inequality for CAT(0) metrics in genus two[END_REF], proved that there exists a piecewise at metric g 0 on the genus two surface Σ 2 which is extremal among all nonpositively curved Riemannian metrics. is metric is composed of six regular octagons. It admits regions where only one systolic loop, i.e., a noncontractible loop of length the systole, passes through every point. Hence this metric cannot be extremal for the general systolic inequality, i.e., without restriction on the curvature. is follows from a result of E. Calabi, cf. [START_REF] Calabi | Extremal isosystolic metrics for compact surfaces, Actes de la Table Ronde de Géométrie Di érentielle[END_REF], which says that at least two systolic loops pass through every point of a systolically extremal surface. In fact, the non-extremality of the metric g 0 for the systolic inequality can also be deduced from a result of S. Sabourau, cf. [START_REF] Sabourau | Systoles des surfaces plates singulières de genre deux[END_REF], which says that no at metric with conical singularities is extremal for the systolic inequality in genus two.

We observe that the piecewise at metric g 0 de ned on the genus two surface Σ 2 may be a potential extremal metric for another systolic problem on Σ 2 . is observation follows from some geometric properties of the surface (Σ 2 , g 0 ). First, it is known that the systolic loops cover a systolically extremal surface. In our case, the 1 -loops and the 2 -loops cover the surface Σ 2 .

Here, by an i -loop we mean a loop of length i (g 0 ) in (Σ 2 , g 0 ) whose free homotopy class lies in Λ i . Second, the unit tangent vectors of these i -loops are well distributed on each tangent plane of the surface. More precisely, their convex hull forms a regular octagon on these tangent planes. In general, the convex hull of the unit tangent vectors of systolic loops on all the known systolically extremal surfaces is symmetric. Finally, an extremal surface of genus at least two tends to have at regions, cf. [START_REF] Calabi | Extremal isosystolic metrics for compact surfaces, Actes de la Table Ronde de Géométrie Di érentielle[END_REF][START_REF] Bryant | On extremals with prescribed lagrangian densities, Manifolds and geometry[END_REF].

In [START_REF] Calabi | Extremal isosystolic metrics for compact surfaces, Actes de la Table Ronde de Géométrie Di érentielle[END_REF], E. Calabi described two piecewise at genus three surfaces and conjectured that one of them is the global minimum for the systolic inequality. Later, S. Sabourau, cf. [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF], proved that these two genus three surfaces are critical for the systolic inequality with respect to slow metric variations. Of course, the systolic volume functional is not necessarily di erentiable and an adequate notion of criticality needs to be introduced. e notion of criticality used in [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF] was introduced in [Na96, EI00] to study the Riemannian surfaces that maximize the product of the area with the rst nonzero eigenvalue of the Laplacian. Other notions of systolically critical metrics were used in [START_REF] Balache | Sur la systole de la sphère au voisinage de la métrique standard[END_REF][START_REF] Balache | A local optimal diastolic inequality on the two-sphere[END_REF][START_REF] Sabourau | Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic[END_REF][START_REF] Paiva | Contact geometry and isosystolic inequalities[END_REF].

In this chapter we show that the metric g 0 on the genus two surface Σ 2 is critical in the sense of [START_REF] Nadirashvili | Berger's isoperimetric problem and minimal immersions of surfaces[END_REF][START_REF] El Sou | Riemannian manifolds admi ing isometric immersions by their rst eigenfunctions[END_REF][START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF] for slow metric variations. More precisely, we prove eorem 3.1.1. Let g t be a slow metric variation of g 0 de ned on the genus two surface Σ 2 . If

1 (g t ) ≥ 1 (g 0 ) and 2 (g t ) ≥ 2 (g 0 ), then area(g t ) ≥ area(g 0 ) + o(t).
e slow metric variations involved in eorem 3.1.1 are analogous to those de ned by S. Sabourau in [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF]. We refer to De nition 3.4.1 for a precise de nition of these metric variations and to the last section of [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF] for examples. Observe for instance that deforming the regular octagons composing the extremal surface (Σ 2 , g 0 ) into non-regular octagons gives rise to a slow metric variation.

With the exception of some recent proofs of systolic inequalities, cf. [Iv02, Sa10], and systolic-like inequalities, cf. [START_REF] Sabourau | Optimal systolic inequalities on Finsler Mobius bands[END_REF], on the projective plane RP 2 , the two-torus T 2 and the Mobius band M, all the other known proofs of the aforementioned optimal geometric inequalities require the uniformization theorem as a main tool. In our proof of eorem 3.1.1, we do not make use of the uniformization theorem. Instead, we rely on recent calibrating methods, cf. [START_REF] Ivanov | On two-dimensional minimal llings[END_REF][START_REF] Ivanov | Filling minimality of Finslerian 2-discs[END_REF][START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF].

Brie y, we embed the universal cover Σ 2 of the genus two surface Σ 2 in an in nite-dimensional space R ∞ as follows. Given that i (g) ≥ i (g 0 ), we construct a 1-Lipschitz equivariant map

Ψ g : Σ 2 → R ∞
using the Busemann functions induced by the li s of the i -loops of Σ 2 , cf. Section 3.3. en, we introduce an appropriate in nitesimally calibrating two-form ω on R ∞ whose pull-back passes to the quotient on the surface Σ 2 . is allows us to show that

Σ 2 (Ψ g ) * ω ≤ c 0 area(Σ 2 , g)
for every Riemannian metric g on Σ 2 , where c 0 is a sharp positive constant. Moreover, the equality holds if g = g 0 . Finally, we prove that if (g t ) t≥0 is a slow metric variation then

Σ 2 (Ψ gt ) * ω - Σ 2 (Ψ g 0 ) * ω = o(t).
is completes the proof, cf. Section 3.4.

An extremal piecewise at metric in genus two

In this section we provide a description for the critical piecewise at genus two surface and we introduce some notations and de nitions. e conformal class of the Bolza surface B admits a piecewise at nonpositively curved Riemannian metric g 0 with 16 conical singularities whose orientation-preserving isometry group is isomorphic to the automorphism group of B. is metric g 0 has been introduced by M. Katz and S. Sabourau in [START_REF] Katz | An optimal systolic inequality for CAT(0) metrics in genus two[END_REF], where they prove that it is extremal for the systolic inequality among all nonpositively curved metrics on the genus two surface Σ 2 . e metric g 0 , de ned up to homothety, on Σ 2 is isometric to the piecewise at genus two surface composed of six identical at regular octagons Ω k with the identi cations given in Figure 3.1. e Weierstrass points of B correspond to the centers of the octagons and the 16 conical singularities are located at their vertices.

Denote by h the length of an edge of a regular octagon Ω k of g 0 . We have By using similar arguments, we calculate 2 (g 0 ). In particular,

area(g 0 ) = 12(1 + √ 2)h 2 . e systole 1 (g 0 ) of the surface (Σ 2 , g 0 ) is computed in [KS06, Lemma 3.2]. Namely, 1 (g 0 ) = 2(1 + √ 2)h.
2 (g 0 ) = 2(2 + √ 2)h.
Recall that an i -loop is a loop in (Σ 2 , g 0 ) of length i (g 0 ) and whose free homotopy class lies in Λ i , cf. (3.1.2) and (3.1.3). We de ne an i -band as follows.

De nition 3.2.1. Let α be a free homotopy class in Λ i . e i -loops in α are parallel to each other and form a at cylinder B α of height h if i = 1 and √ 2 2 h if i = 2. Such a cylinder B α will be called an i -band of (Σ 2 , g 0 ). e soul of an i -band B α is the i -loop of B α equidistant from the boundary components of the i -band. e intersections of the i -bands decompose the surface Σ 2 into 150 polygons, cf. Figure 3.2. ese polygons de ne four regions R 1 , ..., R 4 depending on the number of i -loops that pass through every point in their interior. More precisely, such a polygon ∆ k lies in a region R k if through every point in its interior pass exactly 2k oriented 1 -loops and 8 -2k oriented 2loops, cf. Figure 3.2. Hence, every region R k with k = 4 is composed of exactly 48 identical polygons ∆ k (with 8 polygons in each octagon), while the region R 4 is composed of only 6 identical polygons ∆ 4 . Observe that the polygons ∆ 1 , ∆ 2 , ∆ 3 and ∆ 4 have the shape of a right isosceles triangle, a kite, a right isosceles triangle and a (small) octagon. Furthermore, the right isosceles triangles ∆ 1 are the only polygons which have an edge in common with the six regular at octagons forming the critical surface (Σ 2 , g 0 ), namely their hypothenuses coincide with the edges of the regular octagons Ω k .

Fix an orientation on Σ 2 . Let Ω be one of the six regular octagons composing the surface (Σ 2 , g 0 ).

Denote by ( Σ 2 , g 0 ) the universal cover of (Σ 2 , g 0 ). Let Ω be a li of Ω in the universal cover Σ 2 and let E be an (oriented) edge of the regular octagon Ω. Moreover, let ∆ k be a li of the polygon ∆ k in Σ 2 . e souls of the i -bands li to Σ 2 as follows.

De nition 3.2.2. An 1 -direction of ( Σ 2 , g 0 ) is a g 0 -unit vector ζ = ζ E Ω based at the center of Ω and pointing in the same direction as E, cf. In what follows, we will replace the symbols ζ and ξ by ν when there is no need to distinguish between an 1 -direction and an 2 -direction. Moreover, the indices Ω and E in ν E Ω will be omi ed.

Busemann functions and calibrating forms

In this section, we introduce some notations and preliminary results that will be useful in our proof of eorem 3.1.1. De nition 3.3.1. Let ν be an i -direction. Extend ν to a map, still denoted by ν : R → U g 0 Σ 2 de ned as ν(s) = c ν (s). In particular, ν(0) = ν.

Along with the metric g 0 , consider another Riemannian metric g de ned on Σ 2 and denote by g its li to Σ 2 . where c ν is the g 0 -geodesic line induced by ν and c ν is its projection to Σ 2 .

Here, g ( c ν ) denotes the smallest length of a noncontractible loop in the free homotopy class c ν of c ν with respect to the Riemannian metric g. Proof. By the triangle inequality, we obtain h(t) ≥ d g( c ν (0), c ν (t)) i (g 0 ) -t g ( c ν ) -d g(x, c ν (0)) i (g 0 ).

(3.3.1)

Let β be a length-minimizing loop of (Σ 2 , g) in the homotopy class of c ν , that is, g (β) = g ( c ν ).

Let [t] i be the integer part of t i (g 0 ) , that is,

[t] i = max{k ∈ N | t ≥ k i (g 0 )}.
We have

[t] i i (g 0 ) -t ≥i (g 0 ).

(3.3.2)

Since the genus two surface is orientable, the [t] i -iterate of β is length-minimizing in its homotopy class with respect to g, cf. [START_REF] Gromov | Metric structures for Riemannian and non-Riemannian spaces[END_REF]. Let γ in Σ 2 be a length-minimizing arc connecting c ν (0)

to c ν ([t] i ). Its projection γ to Σ 2 is homotopic to c 3), we deduce that the map h is bounded from below. Now, we show that the map h is bounded from above. Indeed, by the triangle inequality, we derive that

h(t) ≤ d g (x, c ν (0)) + d g ( c ν (0), c ν ([t] i )) + d g ( c ν ([t] i ), c ν (t)) -[t] i g ( c ν ) i (g 0 ).
Moreover,

d g ( c ν (0), c ν ([t] i )) -[t] i g ( c ν ) ≤ 2 diam(Σ 2 , g).
(3.3.4) Indeed, let γ be a g-length minimizing loop in the free homotopy class c ν , that is, g (γ) = g ( c ν ). Let α be a length-minimizing arc joining c ν (0) to γ. We deduce that d g ( c ν (0), c ν ([t] i )) ≤ g (α ∪ γ [t] i ∪ α -1 ) ≤ g (γ [t] i ) + diam(Σ 2 , g).

en, the inequality (3.3.4) is satis ed. Furthermore,

d g ( c ν ([t] i ), c ν (t)) ≤ diam(Σ 2 , g).
is shows that h is bounded from above.

De nition 3.3.5. e map Ψ g ν : Σ 2 → R induced by an i -direction ν, cf. De nitions 3.2.2 and 3.2.3, is de ned as

Ψ g ν (x) = 1 i (g 0 ) g ( c ν ) R/ i (g 0 )Z b g ν(s) (x) -b g -ν(s) (x) ds,
where ν(s) = c ν (s), cf. De nition 3.3.1.

Example 3.3.6. If g = g 0 , the g 0 -gradient ∇ x Ψ g 0 ν is a g 0 -unit tangent vector parallel to and pointing in the same direction as the i -direction ν.

Proposition 3.3.7. Let ν be an i -direction. e map Ψ g ν satis es the following properties:

1. It is di erentiable almost everywhere.

2. Ψ g -ν = -Ψ g ν .

3. e di erential of the map Ψ g ν depends only on the oriented i -band B generated by the vector ν.

Proof. For (1), the map Ψ g ν is 1-Lipschitz for the supremum norm as the Busemann function b g ν is i (g 0 )-Lipschitz and i (g) ≥ i (g 0 ). Hence, it is di erentiable almost everywhere by the for some integer k with n = 2 if i = 1 and n = 4 if i = 2. We derive easily from De nition 3.3.5 that dΨ g η = dΨ g ν .

is completes the proof.

Let Γ be the deck transformation group of Σ 2 . e g 0 -geodesic line c ν induces a unique element γ ν in Γ that leaves c ν globally invariant and such that γ ν . c ν (s) = c ν (s + i (g 0 )).

Moreover, we de ne an action of Γ on the set of unit tangent vectors to the geodesic line c ν as γ ν(s) = (γ. c ν ) (s)

where γ ∈ Γ.

We state now the following properties of the map Ψ g ν .

Proposition 3.3.8. Let ν be an i -direction. e map Ψ g ν : Σ 2 → R satis es the following properties:

1. It is equivariant, that is, for every γ in Γ, Ψ g γ ν (γ.x) = Ψ g ν (x).

2. It passes to the quotient by the cyclic subgroup γ ν and induces a map Ψ g ν : Σ 2 / γ ν → R/ i (g 0 )Z de ned as Ψ g ν (γ ν .x) ≡ Ψ g ν (x) mod i (g 0 ).

Proof. Let x ∈ Σ 2 and γ ∈ Γ. Since γ is an isometry and γ. c ν = c γ.ν , we derive g ( c γ ν ) = g ( c ν ).

Moreover, since the distance d g induced by the Riemannian metric g is Γ-invariant, we derive that b g γ ν (γ.x) = b g ν (x). Hence, the desired result

Ψ g γ * ν (γ.x) = Ψ g ν (x).
Now, since γ -1 ν . c ν (t) = c ν (ti (g 0 )), we derive from the Γ-invariance of d g that d g(γ ν .x, c ν (t)) = d g(x, γ -1 ν . c ν (t)) = d g(x, c ν (ti (g 0 ))).

Hence, b g ν (γ ν .x) = lim sup t→∞ i (g 0 )d g(x, c ν (ti (g 0 ))) -t g ( c ν ) = b g ν (x)i (g 0 ) g ( c ν ).

We deduce that Ψ g ν (γ ν .x) = Ψ g ν (x)i (g 0 ).

(3.3.5)

Remark 3.3.9. As a consequence of Proposition 3.3.8 and the relation (3.3.5), we deduce the following couple of points:

1. γ * (dΨ g γ ν ) = dΨ g ν for every γ ∈ Γ,

2. e map Ψ g ν -Ψ g ν passes to the quotient by the cyclic subgroup γ ν and induces a map

Ψ g ν -Ψ g 0 ν : Σ 2 / γ ν → R such that Ψ g ν (γ ν .x) -Ψ g 0 ν (γ ν .x) = Ψ g ν (x) -Ψ g 0 ν (x).
Actually, these two properties hold if we replace Ψ g ν with the Busemann function b g ν .

Let Ω be a regular octagon of the universal cover ( Σ 2 , g 0 ) of the surface (Σ 2 , g 0 ). We order the (oriented) edges (E j ) 1≤j≤8 of Ω with respect to the cyclic order induced by the orientation of Ω by xing an initial edge E 1 . Denote

ν j = ν E j Ω .
We state the following de nition De nition 3.3.10. Let R ∞ = Π Ω R 8 be the in nite product of R 8 where Ω runs over all the regular octagons of ( Σ 2 , g 0 ). Let

Ψ g : Σ 2 → R ∞
be the map de ned as

π Ω • Ψ g | Ω : Ω → R 8
x → (Ψ g ν j (x)) 8 j=1 where π Ω : R ∞ → R 8

is the canonical projection to the factor R 8 of R ∞ corresponding to Ω. e vector ν j is the idirection that is parallel and point in the same direction as the g 0 -unit vector based at x and tangent to the li of an oriented i -loop.

Example 3.3.11. Consider the regular octagon Ω in Figure 3.2. Suppose that the numbering of the edges of ∂ Ω follows a cyclic order starting at E, then we have

Ψ g | ∆ 2 = (Ψ g ζ 1 , Ψ g ζ 2 , Ψ g ξ 3 , Ψ g ξ 4 , Ψ g ζ 5 , Ψ g ζ 6 , Ψ g ξ 7 , Ψ g ξ 8 )
where ∆ 2 is the gray polygon in Figure 3.2.

Remark 3.3.12. Note that the map Ψ g is not well de ned in the li of the boundaries of the polygons ∆ since there is some ambiguity in the choice of ν j .

De nition 3.3.13. Consider the two-form

ω 0 = 8 j=1 dx j ∧ dx j+2
in R 8 . We de ne a two-form ω on Σ 2 as the pull-back of ω 0 by the map where ν j = ν j Ω and all indices j are taken modulo 8. is de nes a two-form on Σ 2 that we still denote by ω.

π Ω • Ψ g
Note that the two-form ω in De nition 3.3.13 is de ned almost everywhere since the maps Ψ g ν are almost everywhere di erentiable, cf. Proposition 3.3.7 (1).

Proposition 3.3.14. e two-form ω passes to the quotient on Σ 2 .

Proof. Let γ ∈ Γ. Observe that γ preserves the octagonal decomposition of the universal cover ( Σ 2 , g 0 ) and its orientation. erefore, for every regular octagon Ω, there exists a permutation σ of {1, • • • , 8} such that

γ -1 (E j Ω ) = E σ(j) γ -1 ( Ω)
.

at is,

γ -1 (ν j Ω ) = ν σ(j) γ -1 ( Ω)
.

Hence, dΨ g γ -1 ν j = dΨ g ν σ(j) .

By the equivariance property of the map Ψ ν j , cf. Remark 3.3.9, we derive

γ * ω = 8 j=1 dΨ g γ -1 ν j ∧ dΨ g γ -1 ν j+2 = 8 j=1
dΨ g ν σ(j) ∧ dΨ g ν σ(j+2) .

Finally, since the permutation σ arises from an orientation-preserving isometry γ, the angle between ∇ x Ψ g ν σ(j) and ∇ x Ψ g ν σ(j+2) is equal to the angle between ∇ γ(x) Ψ g ν j and ∇ γ(x) Ψ g ν j+2 for every x ∈ Σ 2 at which ω is de ned. erefore, σ(j + 2) = σ(j) + 2.

Reindexing the sum in the expression of the two-forms ω 0 and ω, we derive that γ * ω = ω. Remark 3.3.15.

e induced two-form on Σ 2 will be denoted by ω.

  [KS06], or to the set of metrics in a xed conformal class cf. [Bl61, Ba88, Ba92, Ba06]. In 1961, C. Bla er proved, cf. [Bl61] optimal conformal lower bounds on the area of the Mobius band with boundary M in terms of the product of the least lengths of two classes of curves. e rst class consists of the family F of arcs joining two points on the boundary with non-trivial intersection with the soul of the Mobius band. e second class consists of the family G of loops in the homotopy class of a generator of π 1 (M). Let v be the least length of an arc in F and σ be the least length of a loop in G. en, Bla er (in his notations, v = * and σ = 1 ) obtained the following optimal conformal lower bound on the area of M

Figure 1

 1 Figure 1.1: e curves α a θ , γ u and δ v in the Mobius band {(u, v) | |v| ≤ b} of (K, SF b,ω ).

  Nous établissons des inégalités géométriques optimales sur le ruban de Möbius muni d'une métrique de Finsler. Ces inégalités géométriques relient la systole et la hauteur du ruban de Möbius à son volume de Holmes-ompson. Nous en déduisons une inégalité systolique optimale sur la bouteille de Klein munie d'une métrique de Finsler avec des symétries. Nous décrivons également une famille de métriques extremales dans les deux cas.

  [START_REF] Bla Er | Zur Riemannschen Geometrie im Grossen auf dem Möbiusband[END_REF]. e spherical Mobius bands M a are involved in several extremal conformal systolic inequalities for Riemannian metrics. More precisely, we de ne the orientable systole of a Riemannian Mobius band M as the shortest length of a noncontractible orientable loop in M. It will be denoted by sys + (M). Similarly, we de ne the nonorientable systole of M and denote it by sys -(M). Observe that sys(M) = min sys + (M), sys -(M) . Moreover, we de ne v (M) as the minimal length of the arcs joining (u, -a) to (u, a) in the (u, v)-coordinates of M a , which are homotopic with xed endpoints to the projections of the meridians in S a . For instance, sys + (M a ) = 2π cos(a), sys -(M a ) = π and v (M a ) = 2a. Note that the de nition of v relies on conformal data, namely the longitude-latitude coordinates to de ne the endpoints of the arcs involved in the length minimization.

π 2 )

 2 of the equation tan(x) = 2x and M α(a) ∪ C a,α(a) is the Mobius band obtained by a aching a at cylinder C a,α(a) to the spherical Mobius band M a,α(a) along their boundary. Here, the angle α(a) ∈ [b, a] is implicitly given by a nonlinear equation depending on the conformal type a and the at cylinder C a,α(a) is de ned as the product ∂ + S α (a) × [0, sin a -sin α(a)], where ∂ + S α(a) is a boundary component of S α (a). Alternately, C a,α(a) is the Mercator projection of a connected component of S a \ S α(a) to the vertical cylinder generated by ∂S α(a) .

  Figure 2.1: Systolic (in gray) and meridian directions in the unit tangent plane at a point of latitude v in M a .

  Figure 2.1. ese unit tangent vectors of M a are referred to as systolic directions. e unit tangent vectors to the meridians are called meridian directions. Despite the risk of confusion, we will also call great circles of M a the projections of the great circles of S a to M a . e spherical Mobius band M a , which is extremal for some conformal systolic inequalities among Riemannian metrics, cf. (2.4.1), (2.4.2), and (2.4.3), is not extremal among Finsler metrics. Indeed, by slightly perturbing the quadratically convex norm in each tangent plane away from the systolic and meridian directions of the spherical metric, cf.

  Figure 2.1, we can decrease the area of the Mobius band without changing the systole and the height. is shows that any unit tangent vector to an extremal (quadratically convex) Finsler Mobius band is tangent either to a systolic loop or a height arc, cf. De nition 2.3.4. In other words, the unit tangent vectors induced by the systolic loops and the height arcs of an extremal (quadratically convex) Finsler Mobius band ll in its unit tangent bundle.

a. B x b. B * x Figure 2 . 2 :

 x22 Figure 2.2: Unit ball B x of F a and its dual B * x at a point x of latitude v in S a .

Figure 2

 2 Figure 2.3: e extreme great circles c ± passing through a li p of p in S a .

Example 2.5. 4 .

 4 Let λ ∈ [1, +∞).(E.1) e horizontal translation τ of vector π e x is an isometry of the plane R 2 endowed with the sup-norm. e quotient of the strip R × [0, (λ -1) π 2 ] by the isometry group τ generated by τ is a cylinder C. e Finsler mobius band M obtained by gluing a boundary component of C to M Fa along ∂M Fa with a = π 3 , cf. Section 2.4, satis es vol HT (M) = (1 + λ)π, sys(M) = π and h(M) = λπ. See Figure 2.4.(A). a. Extremal. b. Almost extremal.

Figure 2 . 4 :

 24 Figure 2.4: Almost extremal Finsler Mobius bands when h(M) ≥ sys(M).

  Figure 2.4.(B), so that the resulting space is a Finsler Mobius band M with sys(M) = π, h(M) = ν 1 λπ and vol HT (M) = ν 2 (λ + 1)π, where ν 1 , ν 2 > 1 are arbitrarily close to 1. is Finsler Mobius band is almost extremal for the inequality (2.5.1) when λ ≥ 1.

  and the vertical translation (x, y) → (x, y + 2b) with b > 0. e at Klein bo le K = R 2 /G decomposes into two Mobius bands whose souls correspond to the projections of the lines {y = 0} and {y = b}. e boundary of the two Mobius bands agrees with the projection of the line {y = b 2 } (or {y = -b 2 }).

  be a Finsler Klein bo le (with or without symmetry). Denote by M the Finsler Mobius band obtained by cu ing K open along the soul given by the projection of the line {y = b} to K. e proof of the inequality (2.7.1) in eorem 2.7.2 follows by combining the next three lemmas. Lemma 2.7.3. If h(M) ≥ sys(K) then the inequality (2.7.1) holds true.

Chapter 3 A

 3 systolically critical genus two surface Nous démontrons une inégalité systolique critique sur la surface de genre deux. Plus précisement, il est connu que la surface de genre deux admet une métrique Riemannienne plate à singularités coniques qui est extrémale parmi les métriques à courbure nonpositive pour l'inégalité systolique. Nous montrons que ce e métrique est en fait critique pour des variations lentes de métriques, ce e fois-ci sans hypothèse de courbure, pour un autre problème systolique portant sur les longueurs des plus courts lacets non contractiles dans certaines classes d'homotopie libres données. Ces classes d'homotopie correspondent aux lacets systoliques et deux-systoliques de la surface extrémale.

  [START_REF] Keen | An extremal length on a torus[END_REF], the Klein bo le K, cf. [Ba06, EY15], and the Mobius band M, cf. [Pu52, Bl61, Ba06, EY15].

e

  Bolza surface B is a genus two Riemann surface. It is the smooth completion of the smooth a ne algebraic curve y 2 = x 5 -x. (3.2.1) e set {-1, 0, 1, -i, i, ∞} of roots of the polynomial (3.2.1) (including the point at in nity) corresponds to the set of vertices of the regular octahedral triangulation of the Riemann sphere S 2 under the conformal stereographic projection. Hence, these six points can be identi ed with the rami cation points of the rami ed conformal double cover Q : B → S 2 , or equivalently, with the Weierstrass points of B.

Figure 3

 3 Figure 3.1: e critical surface (Σ 2 , g 0 ) is isometric to the to the piecewise at genus two surface composed of six identical regular octagons with the identi cations given in the gure.

  Figure 3.2. e unit vector ζ induces a geodesic line c ζ on ( Σ 2 , g 0 ) such that c ζ (0) = ζ. e projection c ζ of c ζ to (Σ 2 , g 0 ) is a closed geodesic curve which coincides with the soul of the 1 -band B c ζ . Now, denote by x E the point of Ω lying in the perpendicular bisector of E and at distance √ 2 4 h from E. (Recall that h is the side length of Ω.) Note that the point x E lies in the right isosceles triangle ∆ 1 of Ω with hypotenuse E.De nition 3.2.3. An 2 -direction of ( Σ 2 , g 0 ) is a g 0 -unit vector ξ = ξ E Ω based at x E and pointing in the same direction as the edge E, cf.

  Figure 3.2. is unit vector ξ induces a geodesic line c ξ on ( Σ 2 , g 0 ) such that c ξ (0) = ξ. e projection c ξ of c ξ to (Σ 2 , g 0 ) is a closed geodesic curve which coincides with the soul of the 2 -band B c ξ . Remark 3.2.4.

Figure 3

 3 Figure 3.2: A at regular octagon Ω of the metric g 0 .

Remark 3.3. 3 .

 3 Contrary to the original de nition of a Busemann function, cf. [BH99], the line c ν is geodesic for g 0 but not for g. Moreover, the functionh : R → R t → d g(x, c ν (t)) i (g 0 ) -t g ( c ν )which appears in De nition 3.3.2, is not necessarily non-increasing. Proposition 3.3.4. e real function h is bounded.

[t] iν

  and its length satis es the two relationsg (γ) = d g( c ν (0), c ν ([t] i )) and g (γ) ≥ [t] i g ( c ν ). (3.3.3) Combining (3.3.1), (3.3.2) and (3.3.

  Rademacher theorem. Part (2) follows directly from De nition 3.3.5. For (3), let ν and η be two i -directions generating the same oriented i -band. at is, their basepoints lie in the same i -band and the vectors point in the same direction. Recall that ν(s) = c ν (s), cf. De nition 3.3.1, and the same with η. e de nition of i -directions and the assumption on the vectors ν and η imply that η(s) = ν(s + k n i (g 0 ))

  R 8 on every regular octagon Ω. It can be expressed on Ω as ω = 8 j=1 dΨ g ν j ∧ dΨ g ν j+2

Table 1

 1

	.1 summarizes our study of the extremal metrics for the various inequality types relative
	to the conformal type β.

  Optimal systolic inequalities were studied since the mid-twentieth century a er C. Loewner proved in an unpublished work the following result, cf.[START_REF] Katz | Systolic geometry and topology[END_REF]. Every Riemannian two-torus T 2

	satis es	area(T 2 ) ≥	√ 2 3	sys 2 (T 2 )	(2.1.1)

with equality if and only if T 2 is a at hexagonal torus. Recall that the systole of a nonsimply connected Riemannian surface M , denoted by sys(M ), represents the length of the shortest noncontractible loop of M . is inequality leads us to introduce the systolic area of M de ned as σ R (M ) := inf g area(M, g) sys 2 (M, g) .

(2.1.2)

  [START_REF] Bao | An Introduction to Riemann-Finsler Geometry[END_REF] eorem 1.2.2]. Hence, F induces a strictly convex norm F x on each tangent space T x M with x ∈ M . More speci cally, it gives rise to a Minkowski norm F x on each tangent symmetric space T = inf γ (γ), where the in mum is taken over all piecewise smooth curves γ joining x to y. A geodesic is a curve which locally minimizes length. It is a critical point of the energy functional γ → F 2 (γ(t))dt

	b
	(γ) :=	F ( γ(t))dt.
	a
	By condition (2), (γ) does not depend on the parametrization of γ. Moreover, the functional
	gives rise to a distance function d (here the quadratic convexity condition (3) is necessary).

x M . Working with quadratically convex norms and not merely (strictly) convex norms provides nice dynamical features such as a geodesic ow and a Legendre transform, cf. [Be78, §1]. As in the Riemannian case, notions of length, distance, and geodesics extend to Finsler geometry. Let γ : [a, b] → M be a piecewise smooth curve. e length of γ is de ned as F : M × M → [0, ∞) de ned as d F (x, y)

For x ∈ M , we denote by B x the unit ball of the Minkowski norm F x . Furthermore, the norm F x is Euclidean if and only if B x is an ellipsoid. e dual of B x is de ned as

B * x = {f ∈ T * x M | F * x (f ) ≤ 1} where F *

x is the dual norm of F x . Note that B * x identi es with the polar body B

  Equality holds if T 2 is homothetic to the quotient of R 2 , endowed with a parallelogram norm ||.||, by a la ice generated by two vectors of lengths a and b, parallel to the sides of the unit ball of ||.||.

	vol HT (T 2 ) ≥	2 π	ab.

Keen proved the following. Proposition 2.3.1 [Ke67], [Ka07] §6.2. Let T 2 be a Riemannian two-torus. ere exist two closed curves of lengths a and b generating the rst integral homology group of T 2 such that

vol HT (T 2 ) ≥ √ 3 2 ab.

Equality holds if and only if T 2 is homothetic to the at torus obtained as the quotient of R 2 by a hexagonal la ice.

e proof of Proposition 2.3.1 relies on the uniformization theorem and the Cauchy-Schwarz inequality.

A Finsler version of Proposition 2.3.1 is given by the following result.

Proposition 2.3.2. Let T 2 be a Finsler two-torus. ere exist two closed curves of lengths a and b generating the rst integral homology group of T 2 such that

  Proposition 2.8.1. Let K be a Finsler Klein bo le. en Every symmetric convex body C ⊂ R n admits a unique ellipsoid E(C) of maximal volume among the ellipsoids contained in C. is ellipsoid, called John's ellipsoid, continuously varies with C for the Hausdor topology. Furthermore, it satis es the double inclusion, cf. [Gru07, Klein bo le K with a Finsler metric F , we de ne a continuous Riemannian metric g on K by replacing the Minkowski norm F x on each tangent space T x K by the inner product induced by the John ellipsoid E(B x ), where B x is the unit ball of F x . e double inclusion (2.8.1) satis ed by E(B x ) implies that 1

					vol HT (K) sys 2 (K)	≥	√ π 2	.
	Proof. Corollary 11.2] for instance,		E(C) ⊂ C ⊂	√	n E(C).	(2.8.1)
	Given a √ 2	√ g ≤ F ≤	√ g. Hence,
	sys(F ) ≤ sys(g) and		1 2	vol(g) ≤ vol HT (F )	(2.8.2)
	From the optimal Riemannian systolic inequality on the Klein bo le [Ba86], we obtain
	vol HT (F ) ≥	1 2	vol(g) ≥	1 2	2 √ π	2	sys 2 (g) ≥	√ π 2	sys 2 (F ).
	Remark 2.8.2. see the proof of [ABT, eorem 4.11]. e naive volume bound in (2.8.2) can be improved into vol(g) ≤ π 2 vol HT (F ), is leads to the be er lower bound 4 √ 2 π 2 in Proposi-
	tion 2.8.1.

(g) = sys(g).

particular, it is geodesic. us, the arc γ, which has the subarc α in common with α ∪ α * , agrees with α ∪ α * . In particular, it is invariant by the soul symmetry. As a result, the arc γ induces a noncontractible loop on K a er identi cation of the points of ∂M under the soul symmetry.

Hence, (α) ≥ sys(K).

Lemma 2.7.5. If K is a Finsler Klein bo le with a soul-switching symmetry then h(M) ≥ sys(K).

Proof. Since the Finsler Klein bo le K has a soul-switching symmetry, we can assume that it is composed of two isometric Finsler Mobius bands M 1 and M 2 . By symmetry, we have h(M) = 2h(M 1 ). Given an arc α of M 1 , we denote by α * the arc of M 2 symmetric to α by the soulswitching symmetry. Let γ be a height arc of M 1 .

is arc decomposes into two subarcs α 1 and β 1 connecting ∂M 1 to the soul of M 1 . e arc η =:

, the union of the arc γ and its symmetric image γ * , has endpoints in ∂M and induces a nontrivial class in π 1 (M, ∂M) of length equal to 2h(M 1 ). Moreover, this arc η induces a noncontractible loop on K a er identi cation of the points of ∂M under the soul-switching symmetry. We conclude that h(M) ≥ sys(K).

Lemma 2.7.6. If K is a Finsler Klein bo le with rotational symmetry then h(M) ≥ sys(K).

Proof. Observe that the rotational symmetries of K leave both M and ∂M invariant. Given an arc α of M, we denote by α θ the arc of M symmetric to α by the rotational symmetry of angle θ.

Let γ be a length-minimizing arc of M parametrized (proportionally to its length) by [0, 1] with endpoints in ∂M inducing a nontrivial class in π 1 (M, ∂M). Note that γ is a geodesic arc of length h(M). By the rst variation formula for Finsler metrics, cf. [START_REF] Shen | Lectures on Finsler geometry[END_REF], the geodesic arc γ is perpendicular to ∂M. It follows that the endpoints γ(0) and γ(1) of γ in ∂M are distinct.

Since the Finsler metric is invariant under rotational symmetry, there exists θ ∈ (0, 2π) such that γ θ (0) = γ(1). Both symmetric arcs γ and γ θ are perpendicular to ∂M. In particular, their tangent vectors at γ θ (0) = γ(1) coincide up to sign. erefore, the geodesic arcs γ and γ θ agree up to reparametrization. More precisely, γ θ (s

erefore, the arc γ projects to a closed curve in K. It follows that the length of γ is at least sys(K).

e following example shows that the inequality (2.7.1) is optimal.

Example 2.7.7. e quotient of R 2 , endowed with the sup-norm, by the isometry group G generated by the glide re ection with parameter b = π 2 , cf. De nition 2.7.1, is a Finsler Klein bo le with soul, soul-switching and rotational symmetries, of area 2π and systole π.

We believe that eorem 2.7.2 holds true for every Finsler Klein bo le (not necessarily invariant under soul, soul-switching or rotational symmetries). More precisely, we state the following conjecture.

Proof of eorem 3.1.1

We brie y restate the statement of eorem 3.1.1. Let g t be a slow metric variation of g 0 de ned on Σ 2 . If 1 (g t ) ≥ 1 (g 0 ) and 2 (g t ) ≥ 2 (g 0 ), then area(g t ) ≥ area(g 0 ) + o(t).

Before proceeding to the proof, let us rst de ne what we mean by a slow metric variation.

De nition 3.4.1. A slow metric variation is a family of Riemannian metrics g t with t ≥ 0 such that for every i -direction ν, the following condition is satis ed

where B ν is a fundamental domain in ( Σ 2 , g 0 ) of the band B ν = B cν .

Note that the map b gt ν -b g 0 ν passes to the quotient by the cyclic subgroup c ν , cf. Remark 3.3.9 (2).

Remark 3.4.2. A metric variation g t should have conical singularities as the Riemannian metric g 0 has conical singularities, cf. Section 3.2.

De nition 3.4.3. e F-functional of the map

introduced in De nition 3.3.10 is de ned as

where ω is the quotient two-form on Σ 2 introduced in Section 3.3. e proof of eorem 3.1.1 follows directly from the next two lemmas.

Lemma 3.4.4. e two-form ω is a calibrating form for the map Ψ g . at is,

with equality for g = g 0 .

Proof. Let x ∈ ∆ be a point at which the maps Ψ g ν j are di erentiable. Let v be a g-unit vector in the tangent plane T x Σ 2 . Since the map Ψ g ν j is 1-Lipschitz, we derive that

e desired inequality directly follows.

Let D denote a fundamental domain of the surface (Σ 2 , g 0 ) in its universal cover made of regular octagons Ω. By De nition 3.4.3, the F-functional of the almost everywhere di erentiable map Ψ g is de ned as

When g = g 0 , the vectors ν j and ν j+2 form an oriented orthonormal basis. e same holds for the covectors d x Ψ g 0 ν j and d x Ψ g 0 ν j+2 from Example 3.3.6. Hence,

where d g 0 represents the area form of the Riemannian metric g 0 . In particular, we derive

a er passing to the quotient.

Remark 3.4.5. An analogous two-form ω along systolic directions could be de ned on the genus three surface to recover the result of [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF] without using the Euclidean metrics.

Lemma 3.4.6. Let g t be a slow metric variation of g 0 . en

where Ω is a regular octagon of ( Σ 2 , g 0 ) of side length h. By de nition, an

We still denote by ν the vector given by the systolic direction a er passing to the quotient.

Recall that the di erential dΨ g ν of the map Ψ g ν depends only on the oriented i -band B generated by the vector ν (or rather the oriented geodesic line c ν ), but not on ν, cf. Proposition 3.3.7 (3).

Moreover, by passing to the quotient, we can rearrange the sum in De nition 3.4.3 to be taken over all the oriented i -bands B of (Σ 2 , g 0 ) as follows

where ν B is an i -direction that generates the i -band B and the vector (ν B ) ⊥ is an 1 -direction or an 2 -direction orthogonal to the boundary ∂B and whose basepoint lies in the same regular octagon Ω as the vector ν B . More precisely, if ν B points in the direction of an edge E j of Ω, then (ν B ) ⊥ points in the direction of the edge E j+2 , where the edges follow the cyclic order on ∂Ω.

In particular, we have

where

and

First, we show that I 1 = 0. Recall that from Proposition 3.3.7(2), we have

and the same holds for Ψ. Moreover, since the oriented angle between the vectors ν B and (ν B ) ⊥ is the same as the one between the vectors -(ν B ) ⊥ and ν B , we can write

a er arranging the oriented bands B in the sum (3.4.2). Moreover, the one-forms dΨ g 0 (ν B ) ⊥ are constant over the bands B. erefore, we derive

Now, let v be a tangent vector to the boundary ∂B. e vector v is parallel to the i -direction ν B .

Moreover, by de nition, the unit vectors ν B and (ν B ) ⊥ form an orthonormal basis. Since the g 0 -gradient of Ψ ν B is well de ned as a real function, cf. Remark 3.3.9 (2). Hence, the integral I 1 vanishes.

Finally, we show that I 2 = 0. is follows directly by de nition of a slow metric variation, cf.

De nition 3.4.1, and the Cauchy-Schwartz inequality.