Stéphane Sabourau
email: stephane.sabourau@u-pec.fr

Éphane St

Sabourau

Macroscopic scalar curvature and local collapsing

Keywords: 2010 Mathematics Subject Classification. Primary 53C23; Secondary 53C20, 51K10 Macroscopic scalar curvature, volume growth, Alexandrov/Urysohn width, collapsing, Margulis constant, exponential growth

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The scalar curvature of a closed Riemannian n-manifold M describes how the volume of infinitesimal balls in M compares to the volume of infinitesimal balls in the Euclidean n-space. More precisely, the volume expansion of a ball of radius r centered at x ∈ M satisfies vol(B(x, r)) = ω n r n 1 -scal(M, x) 6(n + 2)

r 2 + O(r 3 ) (1.1)
as r goes to zero, where scal(M, x) is the scalar curvature of M at x and ω n is the volume of a unit ball in the Euclidean n-space; see [START_REF] Gallot | Riemannian geometry[END_REF]Theorem 3.98]. Understanding the relationship between scalar curvature and the topology of M is a major problem in Riemannian geometry.

In this article, we will be interested in macroscopic versions of the following conjecture attributed to Schoen (which follows from a conjecture of Schoen about the Yamabe invariant of hyperbolic manifolds); see [START_REF] Schoen | Variational theory for the total scalar curvature functional for Riemannian metrics and related topics[END_REF] and [START_REF] Guth | Volumes of balls in large Riemannian manifolds[END_REF]. This conjecture was also stated by Gromov [START_REF] Gromov | Large Riemannian manifolds[END_REF][START_REF] Babenko | Minimal volume entropy of simplicial complexes[END_REF].A] with nonsharp constants.

Equivalently, using (1.1), if vol(M, g) < vol(M, hyp) then there exists x 0 ∈ M such that vol g (B(x 0 , r)) > vol hyp (B(r)) for every r > 0 small enough. This conjecture is true in dimension 2 by the Gauss-Bonnet formula and in dimension 3 from Perelman's work; see [START_REF] Kleiner | Notes on Perelman's papers[END_REF]Proposition 93.10]. In higher dimension, it also holds true for Riemannian metrics close enough to the hyperbolic one, see [5, Corollaire C], or if one replaces scalar curvature with Ricci curvature, see [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF].

Following [START_REF] Guth | Metaphors in systolic geometry[END_REF], this leads us to introduce the following notion. The macroscopic scalar curvature of a closed Riemannian n-manifold M at scale r at x ∈ M , denoted by scal r (M, x), is defined as the unique real s such that vol(B M (x, r)) = vol(B H n s (r)) where x is a lift of x in the universal cover M of M and H n s is the simply-connected n-dimensional space form with constant curvature s. It is more conveniently characterized as follows scal r (M, x) ≤ s if and only if vol(B M (x, r)) ≥ vol(B H n s (r)). For example, the macroscopic scalar curvature of a flat torus at any scale is zero. Note that this property fails if one does not take balls in the universal cover of M , but only in M , in the definition of the macroscopic scalar curvature, as otherwise, it would be positive at a large enough scale. By (1.1), at infinitesimally small scale, we have lim r→0 scal r (M, x) = scal(M, x).

In a different direction, the macroscopic scalar curvature at large enough scale provides information on the exponential growth rate of the volume of balls in the universal cover of M , also known as the volume entropy, a much-studied geometric invariant related to the growth of the fundamental group and the dynamics of the geodesic flow. This leads us to define

V M (r) = sup x∈ M vol(B(x, r))
as the maximal volume of a ball of radius r in the universal cover of M . As explained in [START_REF] Guth | Volumes of balls in large Riemannian manifolds[END_REF] and [START_REF] Guth | Metaphors in systolic geometry[END_REF], the celebrated theorem of Besson, Courtois and Gallot [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF] on the minimal volume entropy provides a macroscopic version of Schoen's conjecture 1.1 at large enough scales. Stated in a way suited for comparison (ignoring its rigidity counterpart), this result takes the following form.

Theorem 1.2 (Besson-Courtois-Gallot [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF]). Let (M, hyp) be a closed hyperbolic n-manifold and let g be another Riemannian metric on M . If vol(M, g) < vol(M, hyp) then there exists r 0 > 0 (depending on g) such that for every r ≥ r 0

V M (r) > V H n (r).
In particular, if scal r (g, x) > scal r (hyp) for every r large enough and every x ∈ M then vol(M, g) > vol(M, hyp).

A version of this theorem was first established by A. Katok [20] in dimension 2 and a nonsharp version was obtained before by Gromov [START_REF] Gromov | Volume and bounded cohomology[END_REF] in every dimension.

In [START_REF] Guth | Volumes of balls in large Riemannian manifolds[END_REF], Guth asks for an estimate on r 0 after proving the following nonsharp macroscopic version of Schoen's conjecture.

Theorem 1.3 (Guth [START_REF] Guth | Volumes of balls in large Riemannian manifolds[END_REF]). Let (M, hyp) be a closed hyperbolic n-manifold and let g be another Riemannian metric on M . Then, for every r ≥ 1, there exists a constant δ n,r > 0 depending only on n and r, such that if vol(M, g) ≤ δ n,r vol(M, hyp) then

V M (r) ≥ V H n (r).
In other words, if scal r (g, x) ≥ scal r (hyp) for every x ∈ M then vol(M, g) ≥ δ n,r vol(M, hyp).

Further volume lower bounds have recently been obtained by Alpert and Funano for hypersurfaces in closed manifolds with macroscopic scalar curvature bounded below as a consequence of this result; see [START_REF] Alpert | Macroscopic scalar curvature and areas of cycles[END_REF] and [START_REF] Alpert | Macroscopic stability and simplicial norms of hypersurfaces[END_REF]. Theorem 1.3 gives relatively better volume estimates for unit balls (that is, for r = 1) than for balls of large radius as the constant δ n,r falls off exponentially or faster with r. In [START_REF] Guth | Volumes of balls in large Riemannian manifolds[END_REF], Guth suggests that one could try to combine the approaches of [START_REF] Gromov | Volume and bounded cohomology[END_REF], [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF] and [START_REF] Guth | Volumes of balls in large Riemannian manifolds[END_REF] to obtain a uniform volume estimate with δ n,r = δ n depending only on n. Such a uniform bound was obtained for surfaces by Karam [START_REF] Karam | Growth of balls in the universal cover of surfaces and graphs[END_REF]. In higher dimension, Balacheff and Karam [START_REF] Balacheff | Schoen conjecture for manifolds with non-zero simplicial volume[END_REF] proved a similar result for negatively curved metrics g using techniques developed in [START_REF] Gromov | Volume and bounded cohomology[END_REF].

In this article, we establish the following result in this direction, following a different approach. See Theorem 3.7, Corollary 3.9 and Corollary 3.10 for more general statements.

Theorem 1.4. Let (M, hyp) be a closed hyperbolic n-manifold and let g be another Riemannian metric on M . Then, there exists a constant δ n > 0 depending only on n, such that if

vol(M, g) ≤ δ n then V M (r) ≥ V H n (r) for every r ≥ 1.
In other words, if scal r (g, x) ≥ scal r (hyp) for some r ≥ 1 and every x ∈ M then vol(M, g) ≥ δ n .

This result provides a uniform estimate on the radius of balls in the universal cover of M whose volume is at least the volume of balls of the same radius in H n . Furthermore, the balls of radius r ≥ 1 of large volume in the universal cover of M can be assumed to be centered around the same point; see Corollary 3.10. Actually, we obtain a more general lower bound on the maximal volume of balls that holds for every r ≥ 0; see Corollary 3.9. However, contrary to Theorem 1.3 and Theorem 1.2, the condition of the volume of M which guarantees the existence of balls of large volume in the universal cover of M does not involve the volume of the hyperbolic metric in M . This leads to a stronger condition for manifolds of large hyperbolic volume. Note that similar versions of Theorem 1.4 hold for more general spaces, including closed manifolds admitting negatively curved metrics. See Corollary 3.9 and Corollary 3.10.

The overall strategy of the proof of Theorem 1.4 is inspired by the approach developed in [START_REF] Sabourau | Small volume of balls, large volume entropy and the Margulis constant[END_REF] and extended in [START_REF] Babenko | Minimal volume entropy of simplicial complexes[END_REF] to prove that the minimal volume entropy of a closed manifold admitting a hyperbolic metric is positive. (As explained above, this result was first obtained in [START_REF] Gromov | Volume and bounded cohomology[END_REF] using bounded cohomology argument, before a sharp version was established in [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF] using the so-called barycenter map.) In [START_REF] Sabourau | Small volume of balls, large volume entropy and the Margulis constant[END_REF], we show that for every closed Riemannian n-manifold admitting a hyperbolic metric, there exist two loops γ 1 and γ 2 based at the same point, of length vol(M, g) 1 n , whose homotopy classes generate a subgroup Γ π 1 (M ) of positive algebraic entropy. The proof of this estimate relies on filling techniques developed in [START_REF] Gromov | Filling Riemannian manifolds[END_REF] to establish systolic inequalities, and more precisely, on a lower bound on the filling radius. A lower bound on the volume entropy of M immediately follows from this estimate by standard comparison argument. Now, in order to derive a lower bound of the volume of balls in the universal cover of M for any radius, and not simply for asymptotically large radius, we need to show that there is sufficiently volume around the basepoint of the two loops γ 1 and γ 2 to distribute it in the universal cover of M under the action of Γ. The approach developed in [START_REF] Sabourau | Small volume of balls, large volume entropy and the Margulis constant[END_REF], based on filling techniques, does not readily provide any information on where the volume is located and whether there is enough volume around the basepoint of the two loops. In a different direction, Guth [START_REF] Guth | Volumes of balls in large Riemannian manifolds[END_REF] shows that the volume in the universal cover of a closed aspherical Riemannian manifold is not too diffuse and that a non-negligeable part of it is contained in a ball of given radius, but this result does not say where such a ball is located1 . Later, building upon the construction of [START_REF] Guth | Volumes of balls in large Riemannian manifolds[END_REF], he extended this result into the following theorem.

Theorem 1.5 (Guth [17]). Let M be a closed Riemannian n-manifold. If every ball of radius R in M has volume at most ε n R n for a sufficiently small constant ε n > 0 then the Alexandrov width of M is at most R. That is, there exists a continuous map from M to a simplicial (n -1)complex P whose fibers have radius at most R.

This theorem was generalized by Liokumovich, Lishak, Nabutovsky and Rotman [START_REF] Liokumovich | Filling metric spaces[END_REF] to metric spaces and Hausdorff content instead of volume. The proof of this result was greatly simplified by Papasoglu [START_REF] Papasoglu | Uryson width and volume[END_REF] using the minimal hypersurface approach of Schoen and Yau [START_REF] Schoen | On the structure of manifolds with positive scalar curvature[END_REF] developed by Guth [14] in a similar context. Recently, Nabutovsky [START_REF] Nabutovsky | Linear bounds for constants in Gromov's systolic inequality and related results[END_REF] further extended this approach improving the constant from an exponential bound to a linear one. Now, one can show that the Alexandrov width of a closed Riemannian n-manifold M admitting a hyperbolic metric is bounded from below (up to a multiplicative constant depending only on n) by the minimal length L such that there exist two loops γ 1 and γ 2 based at the same point, of length at most L, whose homotopy classes generate a free subgroup of π 1 (M ); see Section 3. This estimate can be interpreted as a bound on Margulis' constant; see Definition 3.1. Thus, by Theorem 1.5, there exists a ball of radius 1 2 L in M of volume at least c n L n . As explained above, we would like to show that such a ball B is centered at the basepoint of the two loops γ 1 and γ 2 in order to derive an exponential lower bound on the volume of balls in the universal cover M by taking the translates of B in M under the free subgroup induced by the two loops in homotopy. This does not follow directly from the Alexandrov/Urysohn width estimates of [START_REF] Guth | Volumes of balls in Riemannian manifolds and Uryson width[END_REF], [START_REF] Liokumovich | Filling metric spaces[END_REF], [START_REF] Papasoglu | Uryson width and volume[END_REF] or [START_REF] Nabutovsky | Linear bounds for constants in Gromov's systolic inequality and related results[END_REF]. Instead, we need to extend Theorem 1.5 by allowing the fibers in the definition of the Alexandrov width to have diameters of various size so that they capture the thick part of M in an optimal way. This leads us to introduce the following definition. Definition 1.6. Recall that a compact n-polyhedral length space is a finite simplicial n-complex endowed with an intrinsic metric (or length structure). Let X be a compact n-polyhedral length space and ρ : X → (0, ∞) be a function. The space X ρ-collapses if there exists a continuous map π : X → P to a finite simplicial (n -1)-complex such that every fiber of π lies in a ball B(x, ρ(x)) for some x ∈ X.

Note that the Alexandrov width of a space X that ρ-collapses is bounded by sup ρ. Thus, the bound on the size of the fibers of π : X → P in Definition 1.6 better reflects the shape of M than the corresponding uniform bound on the size of the fibers of the map in the definition of the Alexandrov width; see Theorem 1.5. The collapsing of X can also be interpreted in terms of coverings; see Proposition 2.1.

We have the following general theorem which extends Theorem 1.5 and some of its generalizations in [START_REF] Liokumovich | Filling metric spaces[END_REF], [START_REF] Papasoglu | Uryson width and volume[END_REF] or [START_REF] Nabutovsky | Linear bounds for constants in Gromov's systolic inequality and related results[END_REF]. The point of this theorem is that it allows us to have a better control on where the space X can collapse or not.

Theorem 1.7. Let X be a compact n-polyhedral length space and ρ : X → (0, ∞) be a continuous function. Suppose that the space X does not ρ-collapse. Then there exists x 0 ∈ X such that

H n (B X (x 0 , r)) ≥ c n r n for every r ∈ [0, ρ(x 0 )],
where c n is an explicit positive constant depending only on n.

The proof of this theorem relies on the general approach developed by Papasoglu [START_REF] Papasoglu | Uryson width and volume[END_REF] and recently further improved by Nabutovsky [START_REF] Nabutovsky | Linear bounds for constants in Gromov's systolic inequality and related results[END_REF]. As explained above, this approach can be traced back to works of Guth [START_REF] Guth | Systolic inequalities and minimal hypersurfaces[END_REF] and, before, of Schoen and Yau [START_REF] Schoen | On the structure of manifolds with positive scalar curvature[END_REF]. We will follow the arguments of [START_REF] Nabutovsky | Linear bounds for constants in Gromov's systolic inequality and related results[END_REF]. Now, if ρ is given by the Margulis function, see Definition 3.1, and the space X does not ρ-collapse, we can show that, under some condition of the fundamental group of X, there exist two loops of X based at x 0 , of length ρ(x 0 ), generating a free subgroup in π 1 (X) as desired.

The article is organized as follows. In Section 2, we prove Theorem 1.7. Then, in Section 3, we introduce the Margulis function and derive a general version of Theorem 1.4.

Collapse and volume localization

The goal of this section is to give an interpretation of the collapse of a metric space X in terms of coverings through the following classical result and to prove Theorem 1.7 and related results. Given a subset A ⊆ X, we denote by Ā its closure in X.

Proposition 2.1. Let X be a compact n-polyhedral length space and ρ : X → (0, ∞) be a function. The space X ρ-collapses if and only if there exists a finite covering of X of multiplicity at most n by open subsets U i whose closures Ūi are covered by balls B(x i , ρ(x i )) with x i ∈ X.

Remark 2.2. The statement of the proposition would still hold by requiring only U i to be covered by a ball B(x i , ρ(x i )). Though more technical, the current statement simplifies the proofs of forthcoming results, in particular Lemma 2.6.

Proof of Proposition 2.1. Let π : X → P be a continuous map to a finite simplicial (n -1)complex P such that every fiber of π lies in a ball B(x, ρ(x)) for some x ∈ X. By continuity of π and compactness of the fibers, every point p ∈ P has an open neighborhood V p such that π -1 ( Vp ) lies in a ball B(x, ρ(x)) for some x ∈ X. By compactness of P , we extract from the open covering {V p | p ∈ P } of P a finite covering V. Subdividing P if necessary, we can assume that each open star st(v) ⊆ P of the vertices v of P lies in some open subset V of V. By construction, the closures Ūv of the inverse images

U v = π -1 (st(v)) ⊆ X of
the open stars lie in the same balls B(x, ρ(x)) covering the compact fibers π -1 (v). Furthermore, since P is a finite simplicial (n -1)-complex, the finite covering of P formed of the open stars st(v) ⊆ P has multiplicity at most n. Thus, the subsets U v form a finite covering of X of multiplicity at most n.

Let U = {U i | i = 1, . . . , m} be a finite covering of X of multiplicity at most n by open subsets U i with Ūi ⊆ B(x i , ρ(x i )) for some x i ∈ X. Take a partition of unity {φ i } of X subordinated to {U i }. Consider the map π : X → ∆ m-1 defined by

π(x) = (φ 1 (x), . . . , φ m (x))
in the barycentric coordinates of ∆ m-1 . The nerve N of the covering U is a finite simplicial complex with one vertex v i for each open set U i , where v i 0 , . . . , v i k span a k-simplex of N if and only if the intersection ∩ k j=1 U i j is nonempty. By construction, the dimension of the nerve N is one less than the multiplicity of the covering U. That is, dim N ≤ n -1. Now, we identify in a natural way the vertices {v i } of N with the vertices of ∆ m-1 . With this identification, the nerve N of X lies in ∆ m-1 . Furthermore, the image of π lies in N and every fiber of π : X → N lies in one of the open subsets U i ⊆ B(x i , ρ(x i )). Hence, the space X ρ-collapses.

Let us recall Eilenberg's inequality; see [8, 13.3.1].

Theorem 2.3 (Eilenberg's inequality). Let f : X → Y be a Lipschitz map between separable metric spaces (i.e., compact metric spaces). Then for every A ⊆ X and every 0 ≤ m ≤ n, we have

* Y H n-m (A ∩ f -1 (y)) dH m (y) ≤ v n-m v m v n (Lip f ) m H n (A) (2.1)
where * denotes the upper Lebesgue integral, v n is the volume of the unit ball in R n and H n is the n-dimensional Hausdorff measure.

The following result is the equivalent to [23, Lemma 2.1] in our language.

Lemma 2.4. Let X be a compact 1-polyhedral length space and ρ : X → (0, ∞) be a function. Suppose that for every x ∈ X, there exists a ball B(x, r x ) ⊆ X of radius r x ∈ [0, ρ(x)] such that H 1 (B(x, r x )) < r x . Then the space X ρ-collapses.

Proof. By the Eilenberg inequality (2.1) with m = n = 1, for every x ∈ X, there exists τ x ∈ (0, r x ) such that the sphere S(x, τ x ) ⊆ X is empty. Thus, every connected component of X lies in a ball B(x, ρ(x)) for some x ∈ X.

The proof of Theorem 2.7 rests on the following notion, which generalizes the definition given in [START_REF] Nabutovsky | Linear bounds for constants in Gromov's systolic inequality and related results[END_REF]Definition 2.3]; see also [START_REF] Papasoglu | Uryson width and volume[END_REF]. Definition 2.5. Let X be a compact n-polyhedral length space and ρ :

X → (0, ∞) be a bounded function. A compact (n -1)-subcomplex Y of X ρ-separates X if for every connected component C of X \ Y , there exists x ∈ X (depending on C) such that C ⊆ B X (x, ρ(x)).
The following result is a mere adaptation of [23, Lemma 2.5].

Lemma 2.6. Let X be a compact n-polyhedral length space and ρ : X → (0, ∞) be a function. Let Y be a compact (n-1)-subpolyhedron of X which ρ-separates X. Denote by ρ Y the restriction of ρ to Y . Suppose that the space Y ρ Y -collapses. Then the space X ρ-collapses.

Proof. We follow the argument presented in [START_REF] Nabutovsky | Linear bounds for constants in Gromov's systolic inequality and related results[END_REF]Lemma 2.5]. Recall that the polyhedral space Y is endowed with the length structure induced from X. Let U = {U i } be a finite covering of Y of multiplicity at most n -1 by open subsets U i of Y with Ūi ⊆ B Y (y i , ρ(y i )) for some y i ∈ Y . We can thicken every subset U i into an open subset V i of X with Vi ⊆ B Y (y i , ρ(y i )), without increasing the multiplicity. (Note that B Y (y, r) ⊆ B X (y, r) for every y ∈ Y .) Adding the connected components of X \ Y to the family {V i }, we obtain a covering of X of multiplicity at most n (one more than the multiplicity of U) by open subsets of X whose closures lies in balls B X (x, ρ(x)) with x ∈ X. Hence, the space X ρ-collapses.

Denote by λ n = vn

2v n-1 the coefficient in Eilenberg's inequality with m = 1; see (2.1). Note that λ 1 = 1.

Theorem 2.7. Let X be a compact n-polyhedral length space and ρ : X → (0, ∞) be a continuous function. Suppose that the space X does not ρ-collapse. Then there exists x 0 ∈ X such that

H n (B X (x 0 , r)) ≥ c n r n for every r ∈ [0, ρ(x 0 )], where c n = 1 n! n i=1 λ i .
Proof. Let η > 0 be small enough. We can η-approximate the distance function d X (x 0 , •) to x 0 by a piecewise smooth (or even piecewise linear, after taking sufficiently fine subdivisions) (1 + η)-Lipschitz function whose level sets are (n -1)-subpolyhedra. To avoid burdening the argument with standard approximations, we will assume that the spheres S X (x 0 , r) are (n -1)subpolyhedra, keeping in mind that the inequalities below, including the Eilenberg inequality, only hold with an extra 1 + o(η) factor with η > 0 arbitrarily small. Letting η go to zero, this will not change the final estimate.

Let us show by induction on the dimension n of X that for every ε ∈ (0, 1) and every α ∈ (0, 1), there exists x 0 ∈ X such that

H n (B X (x 0 , r)) ≥ 1 -ε n! n i=1 λ i r n (2.2)
for every r ∈ [α ρ(x 0 ), ρ(x 0 )]. The case n = 1 with c 1 = 1 is covered by Lemma 2.4.

In the general case n ≥ 2, fix ε ∈ (0, 1) and α ∈ (0, 1). Denote ε = ε 4 and α = ( ε 2 )

1 n α. Consider a compact (n -1)-subcomplex Y ⊆ X which ρ-separates X and has minimal volume up to δ, where δ = ε (n-1)! ( n i=1 λ i ) α n-1 (inf ρ) n-1 > 0. By Lemma 2.6, the space Y does not ρ Y -collapse. Thus, by induction (replacing ε with ε and α with α ), there exists y 0 ∈ Y such that

H n-1 (B Y (y 0 , s)) ≥ 1 -ε (n -1)! n-1 i=1 λ i s n-1 (2.3)
for every s ∈ [α ρ(y 0 ), ρ(y 0 )]. Denote x 0 := y 0 and let s ∈ [α ρ(x 0 ), ρ(x 0 )). Consider the compact (n-1)-subcomplex Z ⊆ X obtained from Y by replacing Y ∩ B X (y 0 , s) with the sphere S X (y 0 , s). The resulting space Z ρ-separates X. Indeed, the closure of the connected component B X (x 0 , s) of X \ Z is clearly covered by the ball B X (x 0 , ρ(x 0 )). Furthermore, all the other connected components of X \ Z are contained in connected components of X \ Y and therefore their closures also lie in balls B X (x, ρ(x)) with x ∈ X. Now, since Y has minimal volume up to δ among all compact (n -1)-subpolyhedra that ρ-separate X, we derive

H n-1 (Z) ≥ H n-1 (Y ) -δ.
This implies

H n-1 (S X (x 0 , s)) ≥ H n-1 (Y ∩ B X (y 0 , s)) -δ ≥ H n-1 (B Y (y 0 , s)) -δ ≥ 1 -ε (n -1)! n-1 i=1 λ i s n-1 - ε (n -1)! n-1 i=1 λ i s n-1 ≥ 1 -2ε (n -1)! n-1 i=1 λ i s n-1 .
where the second inequality follows from the inclusion B Y (y 0 , s) ⊆ Y ∩ B X (y 0 , s) and the third inequality follows from (2.3) and the bound s ≥ α (inf ρ). Thus, by the Eilenberg inequality (2.1), we obtain

H n (B X (x 0 , r)) ≥ λ n r * α ρ(x 0 ) H n-1 (S X (x 0 , s)) ds ≥ 1 -2ε n! n i=1 λ i r n -α n ρ(x 0 ) n for every r ∈ [α ρ(x 0 ), ρ(x 0 )]. (Note that r ≥ α ρ(x 0 ) since α ≤ α.)
We want this expression to be greater or equal to

1-ε n! ( n i=1 λ i ) r n for every r ∈ [α ρ(x 0 ), ρ(x 0 )]. That is, we want (ε -2ε )r n ≥ α n ρ(x 0 ) n . Since r ≥ α ρ(x 0 ), it is enough to have (ε -2ε )α n ≥ α n .
Replacing ε and α with their expressions in terms of ε and α, we observe that this inequality is satisfied, and in fact, is an equality. Hence,

H n (B X (x 0 , r)) ≥ 1 -ε n! n i=1 λ i r n for every r ∈ [α ρ(x 0 ), ρ(x 0 )].
Take a decreasing sequence (ε m ) converging to zero and fix α = ε m . There exists x m ∈ X such that the n-dimensional Hausdorff measure of B X (x m , s) satisfies a similar lower bound to (2.2) for every s ∈ [ε m ρ(x m ), ρ(x m )] with ε = ε m . By compactness of X, we can assume that (x m ) converges to a point x 0 ∈ X. Also, by compactness of X and continuity of ρ, the function ρ is bounded. Let r ∈ [0, ρ(x 0 )]. By the triangle inequality, we have

B X (x 0 , r) ⊇ B X (x m , r -|x 0 x m |).
Thus, for m large enough, we obtain

H n (B X (x 0 , r)) ≥ 1 -ε m n! n i=1 λ i min{r -|x 0 x m |, ρ(x m )} n .
Since |x 0 x m | goes to zero and ρ is continuous, we obtain the desired lower bound for H n (B X (x 0 , r)).

Remark 2.8. If X is a compact n-polyhderon with a piecewise Riemannian metric (e.g., a closed Riemannian n-manifold), we can apply the coarea formula, see [8, 13.4.2], instead of Eilenberg's inequality in the proof of Theorem 2.7. In this case, we can get rid of the product n i=1 λ i in the expression of c n . Remark 2.9. The lower bound (2.2) holds even if ρ is not continuous, provided it is bounded away from zero.

Let us introduce a couple of definitions: one of topological nature and the other one of geometrical nature. Definition 2.10. Let X be a connected compact n-polyhedral space and φ : π 1 (X) → G be a group homomorphism to a discrete group G. The space X is φ-essential if the classifying map Φ : X → K(G, 1) induced by φ is not homotopic to a continuous map X → P → K(G, 1) which factors out through a simplicial (n -1)-complex P . Definition 2.11. Let X be a connected compact n-polyhedral length space and φ : π 1 (X) → G be a nontrivial group homomorphism to a discrete group G. The φ-systolic function of X is the function sys φ : X → (0, ∞) defined as

sys φ (X, x) = inf{length(γ) | γ is a loop of X based at x with φ([γ]) nontrivial}.
Clearly, the φ-systolic function is continuous and even 2-Lipschitz.

We have the following non-collapsing result, which is a mere generalization of the inequality (4) in [START_REF] Nabutovsky | Linear bounds for constants in Gromov's systolic inequality and related results[END_REF].

Proposition 2.12. Let X be a φ-essential connected compact n-polyhedral length space, where φ : π 1 (X) → G is a nontrivial group homomorphism to a discrete group G. Then the space X does not 1 2 sys φ -collapse. Proof. We follow again an argument presented in [START_REF] Nabutovsky | Linear bounds for constants in Gromov's systolic inequality and related results[END_REF] (suggested by R. Karasev). Suppose that the space X 1 2 sys φ -collapses. Denote by π : X → X the cover corresponding to the subgroup ker φ ¡ π 1 (X). By Proposition 2.1, there exists a finite covering of X of multiplicity at most n by open subsets U i with Ūi ⊆ B(x i , 1 2 sys φ (x i )) for some x i ∈ X. Without loss of generality, we can assume that the open subsets are connected. Every loop γ in Ūi is homotopic to the concatenation of loops of length less than sys φ (x i ) based at x i . (The loops are of the form x i p s ∪ p s p s+1 ∪ p s+1 x i , where the points {p i } are given by a sufficiently fine subdivision of γ.) By definition of the φ-systolic function, each homomorphism π 1 (U i ) → π 1 (X) → G is trivial, where the first homomorphism is induced by the inclusion U i → X. It follows that the preimage π -1 (U i ) of each open set U i decomposes as the disjoint union of open sets α • Ûi ⊆ X, where Ûi is a lift of U i in X and α runs over H = π 1 (X)/ ker φ. Consider the nerve N of the covering {U i } of X and the nerve N of the covering {α • Ûi } of X. We have the following commutative diagram X N

X N

where the horizontal maps correspond to the natural maps from a space to its nerve and the vertical maps are the quotient maps under the natural free actions of the subgroup H G. We can construct an equivariant map N → K to the universal cover K of K = K(G, 1). By construction, this map passes to the quotient giving rise to a map N → K such that the composite X → N → K is homotopic to the classifying map induced by φ. Since the covering {U i } has multiplicity at most n, the nerve of N is a simplicial complex of dimension at most n -1. Therefore, the space X is not φ-essential.

Margulis' constant and volume of large balls

In this section, we introduce the Margulis function and apply the results of the previous section to obtain an exponential lower bound on the maximal volume of balls in the universal cover of a metric with negative curvature, and more generally, of a metric space satisfying similar features.

We will need the following definition of the Margulis function. Definition 3.1. Let X be a compact n-polyhedral length space. Consider a group homomorphism φ : π 1 (X) → G to a discrete group G. For every x ∈ X and µ > 0, denote by

Γ µ φ,x = φ([γ]) ∈ G | γ loop of X based at
x with length(γ) ≤ µ the subgroup of G generated by the φ-image of the homotopy classes of the loops of X based at x of length at most µ.

The Margulis function of X is the function µ φ : X → (0, ∞] defined as

µ φ (x) = sup{µ | Γ µ φ,
x has subexponential growth . Clearly, the Margulis function is continuous and even 2-Lipschitz. With this definition, the subgroup Γ µ φ,x has subexponential growth for every µ < µ φ (x). Note that if the image of φ has exponential growth, the Margulis function is bounded by 2 diam(X). When φ is the identity homomorphism, we simply write µ for µ φ . Remark 3.2. One could replace "has subexponential growth" with "is virtually nilpotent" in the definition of the Margulis function since we will be primarily interested in groups satisfying the Tits alternative; see Definition 3.5.

We have the following non-collapsing result. Proposition 3.3. Every closed n-manifold M with nonzero simplicial volume does not 1 2 µcollapse.

Proof. Suppose that M 1 2 µ-collapses. By Proposition 2.1, there exists a finite covering of M of multiplicity at most n by open subsets U i with Ūi ⊆ B(x i , 1 2 µ(x i )) for some x i ∈ X. Without loss of generality, we can assume that the open subsets U i are connected. The image of π 1 (U i ) under the π 1 -homomorphism induced by the inclusion map U i → M is generated by the homotopy classes of loops of length at most µ(x i ) based at x i . By definition of the Margulis function, see Definition 3.1, this subgroup of π 1 (X) has subexponential growth. Thus, it is amenable. By Gromov's vanishing simplicial volume theorem, see [START_REF] Gromov | Volume and bounded cohomology[END_REF] or [START_REF] Ivanov | Notes on the bounded cohomology theory[END_REF], it follows that M has zero simplicial volume.

Remark 3.4. More generally, every closed n-manifold with nonzero minimal volume entropy does not 1 2 µ-collapse; see [START_REF] Babenko | Minimal volume entropy of simplicial complexes[END_REF]Corollary 2.11]. Let us introduce a quantitative version of the Tits alternative for groups; see [START_REF] Breuillard | Uniform independence in linear groups[END_REF] for earlier results about this notion. Definition 3.5. Let κ be a positive integer. A group G satisfies the κ-Tits alternative if for every symmetric subset S of G containing the identity, either S generates a subgroup of subexponential growth, or there exist two elements in S κ generating a nonabelian free subgroup. In the latter case, the subgroup generated by S has exponential growth and its algebraic entropy (or exponential growth rate) with respect to S is at least log (3) κ . Example 3.6. Our main source of examples of groups satisfying the κ-Tits alternative is given by fundamental groups of closed Riemanian n-manifolds with sectional curvature lying between -k 2 and -1 for some k ≥ 1, where κ = κ(n, k) depends only on n and k; see [START_REF] Dey | Ping-pong in Hadamard manifolds[END_REF].

We can now state the main general result of this section. Theorem 3.7. Let X be a compact n-polyhedral length space and φ : π 1 (X) → G be a group homomorphism whose image satisfies the κ-Tits alternative. Suppose that X does not 1 2 µ φcollapse. Then there exists x0 ∈ X such that

H n (B X (x 0 , r)) ≥ vol n (B H n (C r))
for every r ≥ 0, where C = C n,κ • min{1, 1 sup µ φ } with C n,κ > 0 depending only on n and κ.

Proof. According to Theorem 2.7, there exists x 0 ∈ X such that for every lift x0 ∈ X of x 0 ,

H n (B X (x 0 , r)) ≥ H n (B X (x 0 , r)) ≥ c n r n (3.1)
for every r ∈ [0, 1 2 µ φ (x 0 )], where c n = 1 n! n i=1 λ i . For simplicity, denote µ 0 = µ φ (x 0 ). Let us establish an exponential lower bound for the volume of balls of large enough radius centered at x0 . Lemma 3.8. For every r ≥ 1 2 µ 0 , we have

H n (B X (x 0 , r)) ≥ c n 2 n µ n 0 3 r- µ 0 2 2κµ 0 -1 (3.2)
where • is the floor function.

Proof. The idea of the proof is to find a nonabelian free subgroup F of π 1 (X, x 0 ) generated by loops of length at most 2κµ 0 such that the translates of B X (x 0 , µ 0 2 ) under the free action of F on X are disjoint.

Denote by S the symmetric generating set of Γ 2µ 0 φ,x 0 G formed of the elements φ([γ]) where γ is a loop of X based at x 0 of length at most 2µ 0 . By definition of the Margulis function, see Definition 3.1, the subgroup Γ 2µ 0 φ,x 0 G generated by S has exponential growth. (One could consider Γ µ 0 +ε φ,x 0 instead of Γ 2µ 0 φ,x 0 if one wanted to be more precise.) Note also that every element of S κ is the φ-image of the homotopy class of a loop of X based at x 0 of length at most 2κµ 0 . Thus, since the image of φ in G satisfies the κ-Tits alternative, there exist two loops γ 1 and γ 2 based at x 0 of length at most 2κµ 0 whose homotopy classes α 1 and α 2 generate a nonabelian free subgroup of π 1 (X, x 0 ). Consider a nonabelian free subgroup F π 1 (X, x 0 ) generated by two (nontrivial) elements α 1 and α 2 with d X (x 0 , α i • x0 ) ≤ 2κµ 0 such that the orbit F • x0 has a minimal number of points in B X (x 0 , µ 0 ). (The existence of such subgroup is ensured by the previous discussion.) Suppose that the intersection F • x0 ∩ B X (x 0 , µ 0 ) has a point α • x0 different from x0 with α ∈ F . Denote by k the smallest positive integer such that α k • x0 does not lie in B X (x 0 , µ 0 ). We have

d X (x 0 , α k • x0 ) ≤ d X (x 0 , α k-1 • x0 ) + d X (α k-1 • x0 , α k • x0 ) ≤ µ 0 + d X (x 0 , α • x0 ) ≤ 2µ 0
Thus, the displacement of x0 by α k is at most 2κµ 0 . (Note that κ ≥ 1.) Furthermore, either the subgroup generated by α 1 and α k , or the subgroup generated by α 2 and α k is nonabelian free. Denote by F F this nonabelian free subgroup. Since k ≥ 2, the point α • x0 does not lie in the orbit of F . Therefore, the orbit F • x0 of F has fewer elements in B X (x 0 , µ 0 ) than the orbit F • x0 of F , which contradicts the definition of F . Thus, the only point of the orbit F • x0 lying in B X (x 0 , µ 0 ) is x0 . This implies that the translates of B X (x 0 , µ 0 2 ) under the free action of F on X are disjoint.

Let r ≥ 1 2 µ 0 . The ball B X (x 0 , r) contains all the translates of B X (x 0 , µ 0 2 ) under the elements of F = α 1 , α 2 which can be written as words of length at most w = r -µ 0 2 2κµ 0 with the letters α 1 and α 2 . Since the translates of B X (x 0 , µ 0 2 ) under F are disjoint, we derive from (3.1) that

H n (B X (x 0 , r)) ≥ c n µ n 0 2 n (3 w -1) as desired.
Let us resume the proof of Theorem 3.7 using the polynomial lower bound (3.1) and the exponential lower bound (3.2) on the volume of balls in the universal cover of X.

Recall that

vol n (B H n (r)) = σ n-1 r 0 sinh(t) n-1 dt (3.3)
where σ n-1 is the Euclidean volume of the unit sphere S n-1 . We want to show that

H n (B X (x 0 , r)) ≥ vol n (B H n (C r)) (3.4)
for every r ≥ 0, where C > 0 is a metric-independent constant to determine. We will consider the following situations:

(1) If r ≥ 5κµ 0 then the estimate (3.2) yields the following lower bound

H n (B X (x 0 , r)) ≥ c n 2 n µ n 0 3 r- µ 0 2 4κµ 0 -1 . (3.5)
for the second inequality. Since µ 0 ≤ 1 5κ , the expression µ n-1 0 3 1 8κµ 0 is bounded from below by a positive constant depending only on n and κ. Thus, the previous conditions are satisfied if C ≤ C n,κ for some constant C n,κ > 0 depending only on n and κ.

Case 2. Suppose that r ≥ 5κµ 0 and r ≤ 1. By (3.5) and (3.8), the inequality (3.4) holds if

c n 2 n µ n 0 3 r- µ 0 2 4κµ 0 -1 ≥ σ n-1 n sinh(1) n-1 C n r n . (3.11)
Observe that the left-hand side of this inequality divided by r n is bounded from below by a positive constant depending only on n and κ when 5κµ 0 ≤ r ≤ 1. Thus, the condition (3.11) is satisfied if C ≤ C n,κ for some constant C n,κ > 0 depending only on n and κ.

Case 3. Suppose that r ≤ 5κµ 0 and r ≥ 1. By (3.6) and (3.7), the inequality (3.4) holds if

c n 2 n (5κ) n r n ≥ σ n-1 (n -1)2 n-1 e (n-1)Cr -1 .
(3.12)

Since 1 ≤ r ≤ 5κµ 0 , the left-hand side of this inequality is bounded from below by cn 2 n (5κ) n and the right-hand side is bounded from above by σ n-1

(n-1)2 n-1 e 5(n-1)κµ 0 C -1 . Thus, the condition (3.12) is satisfied if C ≤ Cn,κ µ 0 for some constant C n,κ > 0 depending only on n and κ. Case 4. Suppose that r ≤ 5κµ 0 and r ≤ 1. By (3.6) and (3.8), the inequality (3.4)

holds if c n 2 n (5κ) n r n ≥ σ n-1 n sinh(1) n-1 C n r n . (3.13)
This condition is satisfied if C ≤ C n,κ for some constant C n,κ > 0 depending only on n and κ.

In conclusion, there exists a positive constant C n,κ depending only on n and κ such that for every C ≤ C n,κ • min{1, 1 µ 0 } H n (B X (x 0 , r)) ≥ vol n (B H n (C r)) (3.14) for every r ≥ 0.

We immediately derive the following two corollaries. The first corollary claims that when the volume of X is small enough, the constant C in Theorem 3.7 does not depend on the Margulis function, but only on n and κ. Corollary 3.9. Let X be a compact n-polyhedral length space and φ : π 1 (X) → G be a group homomorphism whose image satisfies the κ-Tits alternative. Suppose that X does not 1 2 µ φcollapse. There exists a constant δ n > 0 such that if H n (X) ≤ δ n then there exists x0 ∈ X such that H n (B X (x 0 , r)) ≥ vol n (B H n (C n,κ r)) for every r ≥ 0, where C n,κ > 0 is a constant depending only on n and κ.

In particular, there exists constant δ n,κ > 0 such that if H n (X) ≤ δ n,κ then there exists x0 ∈ X such that H n (B X (x 0 , r)) ≥ vol n (B H n (r)) for every r ≥ 1.

Proof. Applying the second inequality of (3.1) to r = 1 2 µ φ (x 0 ), we obtain µ φ (x 0 ) n ≤ 2 n c n H n (X).

(3.15)

Thus, if H n (X) ≤ δ n := 1 2 n cn then µ φ (x 0 ) ≤ 1 and we can take C = C n,κ in (3.14), which proves the first part of the corollary.

Recall that the volume of a ball in H n grows exponentially with its radius. Thus, we can choose λ = λ n,κ > 0 large enough such that vol n (B H n (C n,κ λr)) ≥ λ n vol n (B H n (r)) for every r ≥ 1. Denote by λX the compact n-polyhedral length space obtained by multiplying the metric on X by λ. If δ n,κ = δn λ n then H n (λX) = λ n H n (X) ≤ δ n . Therefore, there exists x0 ∈ λX = λ X such that H n (B λ X (x 0 , t)) ≥ vol n (B H n (C n,κ t)) for every t ≥ 0. Hence, H n (B X (x 0 , r)) = 1 λ n H n (B λ X (x 0 , λr)) ≥ 1 λ n vol n (B H n (C n,κ λr)) ≥ vol n (B H n (r)) for every r ≥ 1.

The second corollary is what occurs when we apply the previous result to a closed manifold admitting a metric with negative sectional curvature.

Corollary 3.10. Let M be a closed n-manifold admitting a metric with sectional curvature lying between -k 2 and -1 for some k ≥ 1. Then for every Riemannian metric on M with vol(M ) ≤ δ n,κ for some constant δ n,κ > 0 depending only on n, there exists x0 ∈ M such that vol(B M (x 0 , r)) ≥ vol(B H n (r)) (3.16) for every r ≥ 1.

Proof. As explained in Example 3.6, the fundamental group of M satisfies the κ-Tits alternative for some constant κ = κ(n, k) depending only on n and k; see [START_REF] Dey | Ping-pong in Hadamard manifolds[END_REF]. Since M is a closed manifold admitting a metric of negative curvature, its simplicial volume is nonzero; see [START_REF] Gromov | Volume and bounded cohomology[END_REF]. By Proposition 3.3, this implies that M does not 1 2 µ-collapse. Therefore, the conclusion of Corollary 3.9 holds for M .

Though we believe we can combine the argument of[START_REF] Sabourau | Small volume of balls, large volume entropy and the Margulis constant[END_REF] with the construction in[START_REF] Guth | Volumes of balls in large Riemannian manifolds[END_REF] to obtain the desired result, we will use different, more elementary, techniques to reach the same conclusion. These techniques also have the advantage to apply to more general spaces than closed manifolds.

Partially supported by the ANR project Min-Max (ANR-19-CE40-0014).

(2) If r ≤ 5κµ 0 then the estimate (3.1) yields the following lower bound H n (B X (x 0 , r)) ≥ c n 2 n (5κ) n r n .

(3.6)

(3) If r ≥ 1 then the formula (3.3) combined with the inequality sinh(t) ≤ e t 2 for t ≥ 0 yields the following upper bound

We will consider four cases analyzing the different combinations of situations, starting with the more technical cases.

Case 1. Suppose that r ≥ 5κµ 0 and r ≥ 1. By (3.5) and (3.7), the inequality (3.4) holds if

(3.9)

Comparing the exponential growth rates of the two sides of the inequality, we want log(3)

Thus, we need to compare f (t) = a e αt + a and g(t) = b e βt + b with a, b > 0 and α ≥ β > 0.

We have f (t) ≥ g(t) for every t ≥ t 0 if f (t 0 ) ≥ g(t 0 ) and f (t 0 ) ≥ g (t 0 ). In our case, we need to check the inequality (3.9) and the inequality

for the first inequality (since the term between parenthesis in the left-hand side of (3.9) is bounded from below by 2 when r = 5κµ 0 ) and

for the second inequality using (3.10). Since µ 0 ≥ 1 5κ , these conditions are satisfied if C ≤ Cn,κ µ 0 for some constant C n,κ > 0 depending only on n and κ.


Antonia Jabbour
email: antonia.jabbour@u-pec.fr

Stéphane Éphane Sabourau
email: stephane.sabourau@u-pec.fr

Sharp upper bounds on the length of the shortest closed geodesic on complete punctured spheres of finite area

Keywords: 2020 Mathematics Subject Classification. Primary 53C23; Secondary 53C22, 53C60, 58E11 shortest closed geodesic, optimal systolic inequalities, extremal metrics, noncompact surfaces, punctured spheres, ramified covers

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

This article deals with universal upper bounds on the length of the shortest closed geodesic on surfaces with a complete Riemannian metric of finite area. The existence of a closed geodesic on a closed surface with a Riemannian metric follows from a minimization process using Ascoli's theorem in the nonsimply connected case and from Birkhoff's minmax principle in the simply connected case. For noncompact surfaces with a complete Riemannian metric, closed geodesics may not exist (the Euclidean plane yields an obvious example). However, it was proved by Thorbergsson [START_REF] Thorbergsson | Closed geodesics on non-compact Riemannian manifolds[END_REF] for surfaces with at least three ends and by Bangert [START_REF] Bangert | Closed geodesics on complete surfaces[END_REF] for surfaces with one or two ends that every noncompact surface Σ with a complete Riemannian metric of finite area has a closed geodesic. This allows us to introduce scg(Σ) = inf{length(γ) | γ is a closed geodesic of Σ}.

Note that in higher dimension the existence of a closed geodesic on a closed Riemannian manifold has been established by Fet and Lyusternik, but whether closed geodesics exist or not on any complete noncompact Riemannian n-manifold of finite volume with n ≥ 3 is an open question; see [START_REF] Burns | Open problems and questions about geodesics[END_REF]Question 2.3.1].

In this article, we are interested in finding good (if possible optimal) curvature-free upper bounds on scg(Σ) for every surface Σ with a complete Riemannian metric of finite area A. For every nonsimply connected closed surface, it was independently proved by Hebda [START_REF] Hebda | Some lower bounds for the area of surfaces[END_REF] and Burago-Zalgaller [START_REF] Burago | Geometric inequalities[END_REF] that scg(Σ) ≤ √ 2 area(Σ). In this case, optimal bounds are only known for the torus (Loewner 1949, unpublished, see [START_REF] Katz | Systolic geometry and topology. With an appendix by Jake P. Solomon[END_REF]), the projective plane [START_REF] Pu | Some inequalities in certain nonorientable Riemannian manifolds[END_REF], [START_REF] Pu | Some inequalities in certain nonorientable Riemannian manifolds[END_REF]) and the Klein bottle [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF], [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF]); see also Section 5 and Section 6 for a brief presentation of the extremal inequalities. For the sphere, it was proved by Croke [START_REF] Croke | Area and the length of the shortest closed geodesic[END_REF] that scg(Σ) ≤ 31 area(Σ). This bound was subsequently improved in [START_REF] Nabutovsky | The length of the shortest closed geodesic on a 2dimensional sphere[END_REF], [START_REF] Sabourau | Filling radius and short closed geodesics of the 2-sphere[END_REF] and [START_REF] Rotman | The length of a shortest closed geodesic and the area of a 2-dimensional sphere[END_REF], where the current best bound with 31 replaced with 4 √ 2 is due to Rotman [START_REF] Rotman | The length of a shortest closed geodesic and the area of a 2-dimensional sphere[END_REF]. It is conjectured that the global maximum for the length of the shortest closed geodesic among Riemannian metrics with fixed area on the sphere is attained by the Calabi-Croke sphere (see [START_REF] Croke | Area and the length of the shortest closed geodesic[END_REF] and [START_REF] Croke | Universal volume bounds in Riemannian manifolds[END_REF]), which would yield scg(S 2 ) ≤ 2 area(S 2 ).

(

Recall that the Calabi-Croke sphere is defined as the piecewise flat sphere with three conical singularities of angle 2π 3 obtained by gluing two copies of an equilateral triangle along their boundaries. Though this conjecture remains wide open, it was proved by Balacheff [START_REF] Balacheff | A local optimal diastolic inequality on the two-sphere[END_REF] that the Calabi-Croke sphere is a local maximum (see also [START_REF] Sabourau | Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic[END_REF] for an alternate proof extending to the Lipschitz distance topology). No conjecture is available for other surfaces, except in genus 3, where Calabi constructed nonpositively curved piecewise flat metrics with systolically extremal-like properties; see [START_REF] Calabi | Extremal isosystolic metrics for compact surfaces[END_REF] and [START_REF] Sabourau | Isosystolic genus three surfaces critical for slow metric variations[END_REF] (and [START_REF] Sabourau | A systolic-like extremal genus two surface[END_REF] for related systolic-like properties in genus 2). Under a nonpositive curvature assumption, extremal systolic inequalities have been established for the genus 2 surface and the connected sum of three projective planes; see [START_REF] Katz | An optimal systolic inequality for CAT(0) metrics in genus two[END_REF] and [START_REF] Katz | Dyck's surfaces, systoles, and capacities[END_REF]. In both cases, the extremal nonpositively curved metrics are piecewise flat with conical singularities. It was later proved that this structure is common to all extremal nonpositively curved surfaces; see [START_REF] Katz | Systolically extremal nonpositively curved surfaces are flat with finitely many singularities[END_REF]. Extremal systolic inequalities in a fixed conformal class have been investigated in relation with closed string field theory; see [START_REF] Headrick | Minimal-area metrics on the Swiss cross and punctured torus[END_REF], [START_REF] Headrick | Convex programs for minimal-area problems[END_REF], [START_REF] Naseer | Extremal isosystolic metrics with multiple bands of crossing geodesics[END_REF] for the most recent contributions and the references therein.

For noncompact surfaces Σ with a complete Riemannian metric of finite area, it was also shown by Croke [START_REF] Croke | Area and the length of the shortest closed geodesic[END_REF] that scg(Σ) ≤ 31 area(Σ) (without any curvature assumption). This bound has recently been improved by Beach and Rotman [START_REF] Beach | The length of the shortest closed geodesic on a surface of finite area[END_REF], where the constant 31 is replaced with 4 √ 2 for surfaces with one puncture and with 2 √ 2 for surfaces with at least two punctures. The authors also conjectured that the optimal bound for a punctured sphere with at most three punctures is the same as that for the sphere; see (1.1).

In this article, we show that this conjecture is true for spheres with exactly three punctures and prove an optimal bound for spheres with four punctures. These are the only new optimal universal upper bounds on the length of the shortest closed geodesic obtained during the almost 35 years since Bavard's inequality on the Klein bottle [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF]. We also improve the best known upper bounds for spheres with a higher number of punctures. More precisely, we have the following.

Theorem 1.1. Let Σ = S 2 \ {x 1 , . . . , x k } be a k-punctured sphere with a complete Riemannian metric of finite area. Then the following holds.

(1) If k = 3 then there exists a noncontractible figure-eight geodesic γ on Σ such that length(γ) < 2

1 2 • 3 1 4 area(Σ). (1.2)
Furthermore, this inequality is optimal.

(2) If k ≥ 4 then there exists a noncontractible closed geodesic γ on Σ such that length(γ) < 2 • 3 -1 4 area(Σ). (1.
3) Furthermore, this inequality is optimal when k = 4.

The extremal metric on the three-punctured sphere in (1.2) is modelled on the Calabi-Croke sphere by attaching three cusps of arbitrarily small area around its singularities. This can be done keeping the curvature nonpositively curved.

The extremal metric on the four-punctured sphere in (1.3) is modelled on the tetrahedral sphere by attaching four cusps of arbitrarily small area around its singularities. Here, the tetrahedral sphere is defined as the piecewise flat sphere with four conical singularities of angle π given by the regular tetrahedron. This can also be done keeping the curvature nonpositively curved.

A version of this theorem holds true for complete Finsler punctured spheres of finite Holmes-Thompson area; see Section 2 for a brief account on reversible and non-necessarily reversible Finsler metrics and the Holmes-Thompson volume.

For reversible Finsler metrics, we have the following. Theorem 1.2. Let Σ = S 2 \ {x 1 , . . . , x k } be a k-punctured sphere with a complete reversible Finsler metric of finite area. Then the following holds.

(1) If k = 3 then there exists a noncontractible figure-eight geodesic γ on Σ such that

length(γ) < 2 -1 2 • 3 1 2 • π 1 2
area(Σ).

(1.4)

(2) If k ≥ 4 then there exists a noncontractible closed geodesic γ on Σ such that length(γ) < π

1 2
area(Σ).

(1.5)

Furthermore, this inequality is optimal when k ∈ {4, . . . , 6}.

Contrary to (1.2), the inequality (1.4) on the three-punctured sphere is not necessarily optimal. The extremal metric on the four-punctured sphere in (1.5) is modelled on the sphere S 2 obtained by gluing two copies of the square [0, 1] × [0, 1] endowed with the 1 -metric along their boundary, with four cusps of arbitrarily small area attached around the four vertices of the squares. Note that the four vertices of the squares and the two centers of the squares are at distance 1 2 scg(S 2 ) = 1 from each other. Similarly, the extremal metrics on the five-and six-punctured spheres in (1.5) are obtained by attaching one or two extra cusps of arbitrarily small area around the centers of the squares.

For non-necessarily reversible Finsler metrics, we have the following.

Theorem 1.3. Let Σ = S 2 \ {x 1 , . . . , x k } be a k-punctured sphere with a complete non-necessarily reversible Finsler metric of finite area. Then the following holds.

(1) If k = 3 then there exists a noncontractible figure-eight geodesic γ on Σ such that length(γ) < 2

1 2 • π 1 2 area(Σ). (1.6)
(2) If k ≥ 4 then there exists a noncontractible closed geodesic γ on Σ such that

length(γ) < 2 • 3 -1 2 • π 1 2
area(Σ).

Contrary to Theorem 1.1 and Theorem 1.2, the inequalities in this theorem are not necessarily optimal. In Section 6, we present further optimal inequalities on punctured tori, punctured projective planes and punctured Klein bottle with a few ends, where the number of ends (interestingly) depends whether the metric is Riemannian, reversible Finsler or non-reversible Finsler. These inequalities immediately follow from the corresponding optimal bounds for the underlying closed surfaces.

The following table gives an approximation of the the constant c # (optimal in some cases) for the inequality

scg(Σ) ≤ c # area(Σ)
where Σ is a k-punctured sphere with a complete metric of finite area in the Riemannian, reversible Finsler and non-necessarily reversible Finsler cases when k = 3 or 4.

c # Riemannian reversible Finsler non-reversible Finsler k=3 2 1 2 • 3 1 4 1.861... 2 -1 2 • 3 1 2 • π 1 2 2.170... 2 1 2 • π 1 2 2.506... k=4 2 • 3 -1 4 1.519... π 1 2 1.772... 2 • 3 -1 2 • π 1 2
2.046... We conclude by proving a roughly asymptotically optimal upper bound on the length of the shortest noncontractible closed geodesic on spheres with a large number of punctures; see Theorem 7.2 for a more general statement for genus g surfaces with k punctures.

Theorem 1.4. Let Σ = S 2 \ {x 1 , . . . , x k } be a k-punctured sphere with a complete Riemannian metric of finite area, where k ≥ 3. Then there exists a noncontractible closed geodesic γ on Σ such that

length(γ) ≤ 4 √ 2 area(Σ) k .
Furthermore, the upper bound is roughly asymptotically optimal in k.

Similar upper bounds hold true both in the reversible and non-necessarily reversible Finsler cases (albeit with a different multiplicative constant).

The proofs of our optimal bounds, namely Theorem 1.1, Theorem 1.2 and Theorem 1.3, do not rely on the conformal length method used in [START_REF] Katz | Systolic geometry and topology. With an appendix by Jake P. Solomon[END_REF], [START_REF] Pu | Some inequalities in certain nonorientable Riemannian manifolds[END_REF] and [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF] to establish optimal systolic inequalities. Instead, we exploit ramified covers from the torus to the sphere (the first one was introduced in [START_REF] Sabourau | Sur quelques problèmes de la géométrie systolique[END_REF] and used in [START_REF] Balacheff | Sur des problèmes de la géométrie systolique[END_REF] and [START_REF] Sabourau | Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic[END_REF] in the same context) to connect the extremal properties of the punctured spheres with the extremal equilateral flat torus in Loewner's systolic inequality; see Theorem 5.3.

Acknowledgment.

The second author would like to thank the Fields Institute and the Department of Mathematics at the University of Toronto, where part of this work was accomplished, for their hospitality.

Finsler metrics and Holmes-Thompson volume

This section aims at introducing the notions of Finsler metrics and Holmes-Thompson volume.

Let us recall the definition of a Finsler metric. Definition 2.1. A Finsler metric on a manifold M is a continuous function F : T M → [0, ∞) on the tangent bundle T M of M which is smooth outside the zero section of T M and whose restriction F x := F |TxM to each tangent space T x M is a (possibly asymmetric) norm, that is, (1) Subadditivity:

F x (u + v) ≤ F x (u) + F x (v) for every u, v ∈ T x M ;
(2) Homogeneity: F x (tu) = tF x (u) for every u ∈ T x M ;

(3) Positive definiteness:

F x (u) > 0 for every nonzero u ∈ T x M . A Finsler metric is reversible if F x (-u) = F x (u) for every x ∈ M and u ∈ T x M .
The length of a piecewise smooth curve γ : [0, 1] → M is defined as

length(γ) = 1 0 F (γ (t)) dt
and the distance between two points x and y in M is the infimal length of a curve γ in M joining x to y.

We will consider the following notion of volume.

Definition 2.2. The Holmes-Thompson volume of an n-dimensional Finsler manifold M is defined as the symplectic volume of its unit co-ball bundle B * M ⊆ T * M divided by the volume n of the Euclidean unit ball in R n . That is,

vol(M ) = 1 n B * M 1 n! ω n M
where ω M is the standard symplectic form on T * M .

The Holmes-Thompson volume of a Finsler manifold is bounded from above by its Hausdorff measure, with equality if and only if the metric is Riemannian; see [START_REF] Durán | A volume comparison theorem for Finsler manifolds[END_REF]. Note also that the Holmes-Thompson volume of a Riemannian manifold agrees with its Riemannian volume.

3. Degree-three ramified cover from the torus onto the Calabi-Croke sphere

Consider the piecewise flat sphere (S 2 , g 0 ) with three conical singularities x 1 , x 2 , x 3 obtained by gluing two copies of a flat unit-side equilateral triangle along their boundaries. The sphere (S 2 , g 0 ) is referred to as the Calabi-Croke sphere.

By the theory of coverings [START_REF] Dubrovin | Modern geometry-methods and applications. Part II. The geometry and topology of manifolds[END_REF], there exist a degree-three cover π 0 : T 2 → S 2 ramified over the three vertices x 1 , x 2 , x 3 of S 2 , and a deck transformation map ρ 0 : T 2 → T 2 fixing only the ramification points of π 0 : T 2 → S 2 with ρ 3 0 = id T 2 and π 0 • ρ 0 = π 0 . The ramified cover π 0 : T 2 → S 2 can also be constructed in a more geometrical way as follows. First, cut the sphere along the two minimizing arcs of g 0 joining x 1 to x 2 and x 1 to x 3 . This yields a parallelogram with all sides of unit length. Then, glue three copies of this parallelogram along the two sides between x 3 and the two copies of x 1 to form a hexagon; see Figure 1. By identifying the opposite sides of this parallelogram, we obtain an equilateral flat torus T 2 . The isometric rotation, defined on the hexagon, centered at x 3 and permuting the parallelograms, passes to the quotient and induces a map ρ 0 : T 2 → T 2 . This map gives rise to a degree-three ramified cover π 0 :

T 2 → S 2 .
Thus, the Calabi-Croke sphere can be described as the quotient of an equilateral flat torus by the deck transformation map ρ 0 :

T 2 → T 2 .
Given a Riemannian metric with conical singularities on S 2 , we will endow T 2 with the metric pulled back by π 0 : T 2 → S 2 and its universal cover R 2 with the metric pulled back by the composite map Remark 3.1. The degree-three ramified cover π 0 : T 2 → S 2 was first introduced in [START_REF] Sabourau | Sur quelques problèmes de la géométrie systolique[END_REF] in relation with extremal properties of the Calabi-Croke sphere regarding the length of the shortest closed geodesic. It was later used in [START_REF] Balacheff | A local optimal diastolic inequality on the two-sphere[END_REF] to show that the Calabi-Croke sphere is a local extremum for the length of the shortest closed geodesic among metrics with fixed area. A different proof which does not require the uniformization theorem, but still makes use of the degree-three ramified cover π 0 : T 2 → S 2 , can be found in [START_REF] Sabourau | Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic[END_REF].

R 2 → T 2 → S 2 . x 3 x 1 x 1 x 1 x 2 x 2 x 2
4. Degree-two ramified cover from the torus onto the tetrahedral sphere

Consider the piecewise flat sphere (S 2 , g 1 ) with four conical singularities x 1 , x 2 , x 3 , x 4 given by the unit-side regular tetrahedron. The sphere (S 2 , g 1 ) is referred to as the tetrahedral sphere.

By the theory of coverings, there exist a degree-two cover π 1 : T 2 → S 2 ramified over the four vertices x 1 , x 2 , x 3 , x 4 of S 2 , and a deck transformation map ρ 1 : T 2 → T 2 fixing only the ramification points of π 1 :

T 2 → S 2 with ρ 2 1 = id T 2 and π 1 • ρ 1 = π 1 .
The ramified cover π 1 : T 2 → S 2 can also be constructed in a more geometrical way as follows. First, cut the sphere along the three minimizing arcs of g 1 joining x 1 to x 2 , x 1 to x 3 and x 1 to x 4 . This yields an equilateral triangle with side length two. Then, glue two copies of this triangle along the side passing through x 4 to form a parallelogram; see Figure 2. By identifying the opposite sides of this parallelogram, we obtain an equilateral flat torus T 2 . The symmetry, defined on the parallelogram, centered at x 4 and switching the two equilateral triangles, passes to the quotient and induces a map ρ 1 : T 2 → T 2 . This map gives rise to a degree-two ramified cover

π 1 : T 2 → S 2 . x 4 x 1 x 1 x 1 x 1 x 2 x 2 x 3 x 3 Figure 2.
Degree-two ramified cover of the tetrahedral sphere Thus, the tetrahedral sphere (S 2 , g 1 ) can be described as the quotient of an equilateral flat torus by the deck transformation map ρ 1 :

T 2 → T 2 .
Given a Riemannian metric with conical singularities on S 2 , we will endow T 2 with the metric pulled back by π 1 : T 2 → S 2 and its universal cover R 2 with the metric pulled back by the composite map

R 2 → T 2 → S 2 .
Since the degree of the Riemannian ramified cover π 1 : T 2 → S 2 is equal to two, we have area(T 2 ) = 2 area(S 2 ). (4.1)

Proof of the main theorem

In this section, we recall some basic results in systolic geometry and prove the main theorem of this article, both in the Riemannian case and in the Finsler case. Definition 5.1. Let M be a surface with a complete (Riemannian or Finsler) metric. The systole of M is defined as

sys(M ) = inf{length(γ) | γ is a noncontractible loop of M }.
When M is closed, the systole is attained by the length of a noncontractible closed geodesic referred to as a systolic loop of M .

We will also need the following extension of the notion of systole. Definition 5.2. Let M be a surface with k punctures and p marked points x 1 , . . . , x p , endowed with a complete (Riemannian or Finsler) metric. A loop of M is admissible if it lies in M = M \ {x 1 , . . . , x p } and is not homotopic in M to a point, a multiple of a noncontractible simple loop of a cusp, or a multiple of some small circle around a marked point x i . The marked homotopy systole of M is the infimal length of the admissible loops of M . It is denoted by sys * (M ).

Let us recall Loewner's systolic inequality in the Riemannian case (unpublished), see [START_REF] Katz | Systolic geometry and topology. With an appendix by Jake P. Solomon[END_REF], in the reversible Finsler case, see [START_REF] Sabourau | Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic[END_REF], and in the nonnecessarily reversible case, see [START_REF] Álvarez Paiva | Isosystolic inequalities for optical hypersurfaces[END_REF]. Theorem 5.3 ([22], [START_REF] Sabourau | Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic[END_REF], [START_REF] Álvarez Paiva | Isosystolic inequalities for optical hypersurfaces[END_REF]). Let T 2 be a torus. Then the following statements hold true.

(1) For every Riemannian metric on T 2 , sys(T 2 ) ≤ 2

1 2 • 3 -1 4 area(T 2 ) (5.1)
with equality if and only if T 2 is an equilateral flat torus.

(2) For every reversible Finsler metric on T 2 ,

sys(T 2 ) ≤ 2 -1 2 • π 1 2 area(T 2 ) (5.2)
with equality if T 2 is a square flat torus endowed with the 1 -or ∞ -metric. (3) For every non-necessarily reversible Finsler metric on T 2 , sys(T 2 ) ≤ 2

1 2 • 3 -1 2 • π 1 2 area(T 2 ) (5.3)
with equality if T 2 is homothetic to the quotient of R 2 , endowed with the non-symmetric norm whose unit disk is the triangle with vertices (1, 0), (0, 1) and (-1, -1), by the lattice Z 2 .

We can now proceed to the proof of the main theorem.

Proof of Theorem 1.1. Consider the case (1). Let Σ be a 3-punctured sphere with a complete Riemannian metric of finite area. Take three cylindrical ends C1 , C2 , C3 of Σ. For every i = 1, . . . , 3, take a cylindrical end

C i ⊆ Ci with d(C i , ∂ Ci ) > 2 1 2 • 3 1 4
area(Σ).

(5.4) Collapse every end C i to a point x i . This gives rise to a sphere S 2 with a Riemannian metric with three singularities x 1 , x 2 , x 3 . Note that area(S 2 ) < area(Σ).

Consider the degree-three ramified cover π 0 : T 2 → S 2 with branched points x 1 , x 2 , x 3 described in Section 3. Denote by p i the preimage of x i under π 0 : T 2 → S 2 for i = 1, . . . , 3. Endow T 2 with the singular pullback Riemannian metric. The metric on T 2 can be smoothed out in the neighborhood of its singularities, keeping the area and the systole fixed. By Loewner's inequality (5.1) and the relation (3.1), there exists a noncontractible closed geodesic γ on T 2 with length(γ) ≤ 2

1 2 • 3 1 4
area(S 2 ).

(5.5)

The loop γ does not pass through any singularity p i . Otherwise, by the distance estimate (5.4) giving a lower bound on the distance between the center p i of the quotient disk D i = Ci /C i and its boundary, and the length upper bound (5.5), the loop γ passing through p i would lie in the topological disk D i of T 2 . This would contradict the noncontractibility of γ in T 2 . Thus, the systolic loop γ projects to a closed geodesic of S 2 \{x 1 , x 2 , x 3 } ⊆ Σ under π 0 : T 2 → S 2 . By [START_REF] Sabourau | Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic[END_REF]Lemma 7.1], this closed geodesic is a figure-eight geodesic of S 2 (with exactly one singularity x i in each of the three domains of S 2 it bounds). This concludes the proof of the case (1) in Theorem 1.1.

In the case (2), the proof is similar. Start with a 4-punctured sphere Σ with a complete Riemannian metric of finite area. Take four cylindrical ends C 1 , C 2 , C 3 , C 4 of Σ of small area located far away from the core of the surface. Collapse the cylindrical ends into points x i . This gives rise to a sphere S 2 with a Riemannian metric with four singularities x i . Consider the degree-two ramified cover π 1 : T 2 → S 2 with branched points x i described in Section 4. By Loewner's inequality (5.1) and the relation (4.1), there exists a noncontractible closed geodesic γ on T 2 with length(γ)

≤ 2 • 3 -1 4 area(S 2 ).
As previously, the systolic loop γ does not pass through a ramification point of π 1 : T 2 → S 2 and projects to a closed geodesic 1 of S 2 \ {x 1 , x 2 , x 3 , x 4 } ⊆ Σ. This concludes the proof of the case (2) in Theorem 1.1.

In both cases, the extremal metrics are described in the introduction right after Theorem 1.1.

Remark 5.4. In the Finsler case, we simply need to replace Loewner's inequality (5.1) with (5.2) for reversible Finsler metrics, and with (5.3) for non-necessarily reversible Finsler metrics. This leads to the Finsler version of the main theorem given by Theorem 1.2 and Theorem 1.3. The only minor novelty is when k = 5 or 6. In this case, we take k cylindrical ends C i of Σ of small area located far away from the core of the surface, and collapse the cylindrical ends into points x i . Consider the degree-two ramified cover π 1 : T 2 → S 2 branched only at four points x 1 , . . . , x 4 as previously. Apply the Finsler version of Loewner's inequality and observe that the systolic loops of T 2 do not pass through the preimages π -1 1 (x i ) of the singularities of S 2 and project to closed geodesics of S 2 \ {x 1 , . . . , x k } ⊆ Σ as required.

Remark 5.5. Contrary to the Riemannian case, the extremal (reversible or non-reversible) Finsler metric on T 2 does not pass to the quotient under the deck transformation groups of π 0 : T 2 → S 2 , which explains why the inequalities (1.4) and (1.6) may not be optimal. The same occurs for the extremal non-reversible Finsler metric on T 2 with the deck transformation group of π 1 : T 2 → S 2 . However, the extremal reversible Finsler metric on T 2 does pass to the quotient under the deck transformation group of π 1 : T 2 → S 2 . In this case, the inequality (1.5) is optimal and the approximating metrics are described in the introduction right after Theorem 1.2.

1 Arguing as in [START_REF] Sabourau | Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic[END_REF]Lemma 7.1], one can show that the systolic loop γ of T 2 projects either to a simple closed geodesic surrounding exactly two branched points of S 2 on each side, or to a figure-eight geodesic with exactly one or two branched points in each of the three domains of S 2 it bounds.

Remark 5.6. One may wonder if our technique can be applied to other ramified covers T 2 → S 2 in order to derive sharp upper bounds on the length of the shortest closed geodesics on other Riemannian punctured spheres Σ. At the heart of the matter is the property that the extremal equilateral flat metric on T 2 should induce an extremal Riemannian metric on Σ but also on S 2 with marked points/branched points x i corresponding to the ends of Σ. In particular, the marked homotopy systole of S 2 should be greater or equal to the systole of T 2 . This implies that the ramification points p i of T 2 must be at distance at least 1 2 sys(T 2 ) from each other. Thus, the open disks D(p i , 1 4 sys(T 2 )) must be disjoint. Since the area of each of these flat disks is equal to π 16 sys(T 2 ) 2 , we deduce that the number of ramification points of T 2 does not exceed area(T 2 )

π 16 sys(T 2 ) 2 = 8 π √ 3 = 4.4...
Therefore, the number of ramification points is at most 4. In conclusion, our method to find extremal Riemannian metrics based on Loewner's inequality on the torus cannot apply to punctured spheres with more than 4 ends.

Extremal metrics on noncompact surfaces

In this section, we present other examples of noncompact surfaces admitting sharp upper bounds on the length of their shortest closed geodesic. The inequality (6.1) is not optimal when k is large, see Theorem 7.2, but it is for small values of k. For this, one needs to find k points on M at distance at least 1 2 sys(M ) from each other. Compare with Remark 5.6. For instance, we can consider the extremal (Riemannian or Finsler) metrics on the torus as follows; see Theorem 5.3. The equilateral flat torus (see Theorem 5.3.( 1)) admits 4 such points; see Figure 3a. Attaching cusps of arbitrarily small area around these 4 points, we construct an almost extremal Riemannian metric on the torus with k punctures, where k ≤ 4. Similarly, the square flat torus with the 1 -metric (see Theorem 5.3.( 2)) admits 8 such points; see Figure 3b. As previously, we can construct an almost extremal reversible Finsler metric on the torus with k punctures, where k ≤ 8. Finally, the square torus with the extremal non-reversible Finsler metric (see Theorem 5.3.(3)) admits 9 points whose distance, back and forth, between any pair of them is at least sys(T 2 ); see Figure 3c. (Note that the asymmetric distance between two of these points might be less than 1 2 sys(T 2 ) but the distance in the opposite direction makes up for it and their sum is at least sys(T 2 ).) As previously, we can construct an almost extremal non-reversible Finsler metric on the torus with k punctures, where k ≤ 9. The same construction applies to the projective plane where the extremal metric is given by the canonical metric both in the Riemannian and Finsler cases; see [START_REF] Pu | Some inequalities in certain nonorientable Riemannian manifolds[END_REF] and [START_REF] Ivanov | On two-dimensional minimal fillings[END_REF]. More precisely, we can construct an almost extremal metric on the projective plane with k punctures, where k ≤ 3.

This construction also applies to the Klein bottle where the extremal metric is known both in the Riemannian and reversible Finsler settings. Specifically, the extremal Riemannian Klein bottle is obtained by attaching along their boundary two copies of the Mobius band defined as the quotient of the π 4 -neighborhood of the equator on the standard sphere by the antipodal map; see [START_REF] Bavard | Inégalité isosystolique pour la bouteille de Klein[END_REF]. While the extremal Finsler Klein bottle is the square flat Klein bottle with the 1 -metric; see [START_REF] Cossarini | Discrete surfaces with length and area and minimal fillings of the circle[END_REF]. Thus, we can construct an almost extremal metric on the Klein bottle with k punctures, where k ≤ 4 in the Riemannian case and k ≤ 8 in the reversible Finsler case.

Surfaces with many punctures

In this section, we show a roughly asymptotically optimal upper bound on the length of the shortest closed geodesic on a surface with a large number of punctures.

We will need the following result, which can be found in [START_REF] Balacheff | Short loop decompositions of surfaces and the geometry of Jacobians[END_REF]Lemma 6.5]. The following result implies Theorem 1.4 when g = 0.

Theorem 7.2. Let Σ be a surface of genus g with k punctures, endowed with a complete Riemannian metric of finite area. Then

sys * (Σ) ≤ C log(g + 2) √ g + k + 1 area(Σ)
where C is an explicit universal constant.

Proof. Take k cylindrical ends C i ⊆ Σ far away from the core of Σ so that

d(C i , C j ) > sys * (Σ) (7.4)
for every i = j, and length(α) > sys * (Σ) (7.5) for every arc α of Σ with endpoints in C i inducing a nontrivial class in π 1 (Σ, C i ).

Collapse every end C i to a point x i . Denote by M the resulting closed surface with k marked points x 1 , . . . , x k . The Riemannian metric on Σ induces a metric on M that can be smoothed out in the neighborhood of the singularities x i , keeping the area and the marked homotopy systole fixed. Note that area(M ) ≤ area(Σ).

Claim 7.3. We have sys * (Σ) ≤ sys(M ).

Proof. Let us show that length(γ) ≥ sys * (Σ) for every noncontractible loop γ of M . By (7.4), we can assume that the loop γ passes through at most one singularity of M , otherwise we are done. We can further assume that the loop γ does not pass through any singularity x i of M . Otherwise, it would admit an arc α ⊆ Σ with endpoints in C i inducing a nontrivial class in π 1 (Σ, C i ) as a lift under the quotient map Σ → M . By (7.5), we would be done. Thus, the loop γ of M also lies in Σ. Furthermore, the loop γ is noncontractible in Σ, even after collapsing the ends of Σ. It follows that γ is an admissible loop of Σ. Therefore, length(γ) ≥ sys * (Σ).

The roughly asymptotically optimal systolic inequality for closed surfaces of large genus [START_REF] Gromov | Filling Riemannian manifolds[END_REF] [START_REF] Gromov | Systoles and intersystolic inequalities[END_REF] (see also [START_REF] Balacheff | Sur des problèmes de la géométrie systolique[END_REF] and [START_REF] Katz | Entropy of systolically extremal surfaces and asymptotic bounds[END_REF] for alternate proofs) applied to M , combined with the relations sys * (Σ) ≤ sys(M ) and area(M ) ≤ area(Σ), shows that sys * (Σ) ≤ C log(g + 2) √ g + 1 area(Σ) (7.6) for some explicit universal constant C . This proves the theorem when k = 0. Now, consider the closed surface M obtained by applying Lemma 7.1 to the closed surface M with its k marked points, with R = 1 4 sys * (M ). Observe that d M (x i , x j ) ≥ 1 2 sys * (M ). Otherwise we could find a figure-eight curve on M of length less than sys * (M ), in the neighborhood of the segment [x i , x j ], surrounding both x i and x j . This would contradict the relation (7. Remark 7.4. Theorem 7.2 extends to Finsler metrics. Indeed, given a non-necessarily reversible Finsler metric F on Σ, we can replace F with a reversible Finsler metric F defined by F (v) = F (v) + F (-v). Then we replace F with the continuous Riemannian metric g whose unit disk agrees with the inner Loewner ellipsoid associated to the unit tangent disk of F . By construction, sys * (Σ, F ) ≤ sys * (Σ, g) and area(Σ, g) ≤ λ area(Σ, F ) for some explicit universal constant λ. (We refer to the proofs of Corollary 4.12 and Theorem 4.13 in [START_REF] Álvarez Paiva | Isosystolic inequalities for optical hypersurfaces[END_REF] for the details.) Thus, we can apply Theorem 7.2 to the Riemannian metric g and immediately derive a similar upper bound on the length of the shortest closed geodesic of F in terms of the Holmes-Thompson area of F (with a different multiplicative constant).

Figure 1 .

 1 Figure 1. Degree-three ramified cover of the Calabi-Croke sphere

Proposition 6 . 1 .

 61 Let M be a closed surface with a systolically extremal (Riemannian or Finsler) metric. Denote by Σ = M \ {x 1 , . . . , x k } the surface M with k punctures. Then every complete (Riemannian or Finsler) metric on Σ satisfies sys * (Σ) ≤ c(M ) area(Σ) (6.1) where c(M ) = sys(M ) √ area(M ).Proof. To prove this upper bound on sys * (Σ), simply collapse small enough and far enough cylindrical ends C i of Σ. The resulting surface M (where the metric is smoothed out) is homeomophic to M and satisfies sys * (Σ) ≤ sys(M ) and area(M ) ≤ area(Σ). Since the metric on M is systolically extremal, we clearly have c(M ) ≤ c(M ) and the desired result immediately follows.

  Riemannian (b) reversible Finsler (c) non-reversible Finsler

Figure 3 .

 3 Figure 3. Separated points on the torus

Lemma 7 . 1 . 3 )

 713 Let M be a closed surface with a Riemannian metric and k marked points x 1 , . . . , x k , with k ≥ 3. Fix R ∈ (0, 1 4 sys * (M )]. Then there exists a closed Riemannian surface M such that area( M ) ≤ area(for every disk D(R) of radius R in M .

10 g+ 1 ≥ 1 10 √

 10110 2). It follows that the open disks D(x i , 1 4 sys * (M )) of M are disjoint. Combined with (7.1), we derive area(Σ) ≥ area( M ) ≥ k i=1 area D(x i , 1 4 sys * (M )). Now, by (7.3), we have area D(x i , 1 4 sys * (M )) ≥ 1 32 sys * (M ) 2 . k ≥ 10 g + 1 log(g + 2) 2 , then k ≥ 1 + k + 1 log(g + 2) 2 and the desired upper bound on sys * (Σ) follows from (7.7). Otherwise, if k ≤ 10 g + 1 log(g + 2) 2 , then √ g g + k + 1 and the desired upper bound follows from (7.6).

Partially supported by the ANR project Min-Max (ANR-19-CE40-0014).