arXiv subjects

Stéphane Sabourau

Publications and source records attributed to Stéphane Sabourau.

14 recordsLinked to original sources

arXiv2025

Complete 3-manifolds of positive scalar curvature with quadratic decay

We prove that if an orientable 3-manifold $M$ admits a complete Riemannian metric whose scalar curvature is positive and has a subquadratic decay at infinity, then it decomposes as a (possibly infinite) connected sum of spherical manifolds and $\mathbb{S}^2 \times \mathbb{S}^1$ summands. This generalises a theorem of Gromov and Wang by using a different, more topological, approach. As a result, the manifold $M$ carries a complete Riemannian metric of uniformly positive scalar curvature, which partially answers a conjecture of Gromov. More generally, the topological decomposition holds without any scalar curvature assumption under a weaker condition on the filling discs of closed curves in the universal cover based on the notion of fill radius. Moreover, the decay rate of the scalar curvature is optimal in this decomposition theorem. Indeed, the manifold $\mathbb{R}^2 \times \mathbb{S}^1$ supports a complete metric of positive scalar curvature with exactly quadratic decay, but does not admit a decomposition as a connected sum.

math.DG↗

arXiv2024

Minimal volume entropy and fiber growth

This article deals with topological assumptions under which the minimal volume entropy of a closed manifold $M$, and more generally of a finite simplicial complex $X$, vanishes or is positive. These topological conditions are expressed in terms of the growth of the fundamental group of the fibers of maps from a given finite simplicial complex $X$ to lower dimensional simplicial complexes $P$. We also give examples of finite simplicial complexes with zero simplicial volume and arbitrarily large minimal volume entropy.

math.GT↗

arXiv2024

Diastolic and isoperimetric inequalities on surfaces

We prove a universal inequality between the diastole, defined using a minimax process on the one-cycle space, and the area of closed Riemannian surfaces. Roughly speaking, we show that any closed Riemannian surface can be swept out by a family of multi-loops whose lengths are bounded in terms of the area of the surface. This diastolic inequality, which relies on an upper bound on Cheeger's constant, yields an effective process to find short closed geodesics on the two-sphere, for instance. We deduce that every Riemannian surface can be decomposed into two domains with the same area such that the length of their boundary is bounded from above in terms of the area of the surface. We also compare various Riemannian invariants on the two-sphere to underline the special role played by the diastole.

math.DG↗

arXiv2023

Minimal area of Finsler disks with minimizing geodesics

We show that the Holmes--Thompson area of every Finsler disk of radius $r$ whose interior geodesics are length-minimizing is at least $\frac{6}π r^2$. Furthermore, we construct examples showing that the inequality is sharp and observe that the equality case is attained by a non-rotationally symmetric metric. This contrasts with Berger's conjecture in the Riemannian case, which asserts that the round hemisphere is extremal. To prove our theorem we discretize the Finsler metric using random geodesics. As an auxiliary result, we show that the integral geometry formulas of Blaschke and Santaló hold on Finsler manifolds with almost no trapped geodesics.

math.MG↗

arXiv2020

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

We establish sharp universal upper bounds on the length of the shortest closed geodesic on a punctured sphere with three or four ends endowed with a complete Riemannian metric of finite area. These sharp curvature-free upper bounds are expressed in terms of the area of the punctured sphere. In both cases, we describe the extremal metrics, which are modeled on the Calabi-Croke sphere or the tetrahedral sphere. We also extend these optimal inequalities for reversible and non-necessarily reversible Finsler metrics. In this setting, we obtain optimal bounds for spheres with a larger number of punctures. Finally, we present a roughly asymptotically optimal upper bound on the length of the shortest closed geodesic for spheres/surfaces with a large number of punctures in terms of the area.

math.DG↗

arXiv2020

Sweepouts of closed Riemannian manifolds

We show that for every closed Riemannian manifold there exists a continuous family of $1$-cycles (defined as finite collections of disjoint closed curves) parametrized by a sphere and sweeping out the whole manifold so that the lengths of all connected closed curves are bounded in terms of the volume (or the diameter) and the dimension $n$ of the manifold, when $n \geq 3$. An alternative form of this result involves a modification of Gromov's definition of waist of sweepouts, where the space of parameters can be any finite polyhedron (and not necessarily a pseudomanifold). We demonstrate that the so-defined polyhedral $1$-dimensional waist of a closed Riemannian manifold is equal to its filling radius up to at most a constant factor. We also establish upper bounds for the polyhedral $1$-waist of some homology classes in terms of the volume or the diameter of the ambient manifold. In addition, we provide generalizations of these results for sweepouts by polyhedra of higher dimension using the homological filling functions. Finally, we demonstrate that the filling radius and the hypersphericity of a closed Riemannian manifold can be arbitrarily far apart.

math.DG↗

arXiv2020

Macroscopic scalar curvature and local collapsing

Consider a closed Riemannian $n$-manifold $M$ admitting a negatively curved Riemannian metric. We show that for every Riemannian metric on $M$ of sufficiently small volume, there is a point in the universal cover of $M$ such that the volume of every ball of radius $r \geq 1$ centered at this point is greater or equal to the volume of the ball of the same radius in the hyperbolic $n$-space. We also give an interpretation of this result in terms of macroscopic scalar curvature. This result, which holds more generally in the context of polyhedral length spaces, is related to a question of Guth. Its proof relies on a generalization of recent progress in metric geometry about the Alexandrov/Urysohn width involving the volume of balls of radius in a certain range with collapsing at different scales.

math.DG↗

arXiv2017

Mixed sectional-Ricci curvature obstructions on tori

We establish new obstruction results to the existence of Riemannian metrics on tori satisfying mixed bounds on both their sectional and Ricci curvatures. More precisely, from Lohkamp's theorem, every torus of dimension at least three admits Riemannian metrics with negative Ricci curvature. We show that the sectional curvature of these metrics cannot be bounded from above by an arbitrarily small positive constant. In particular, if the Ricci curvature of a Riemannian torus is negative, bounded away from zero, then there exist some planar directions in this torus where the sectional curvature is positive, bounded away from zero. All constants are explicit and depend only on the dimension of the torus.

math.DG↗

arXiv2016

Strong deformation retraction of the space of Zoll Finsler projective planes

We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every Zoll Finsler projective plane to the geodesic flow of the round metric through a family of smooth free circle actions induced by the curvature flow of the canonical round projective plane. This construction provides a description of the geodesics of the Zoll Finsler metrics along the retraction.

math.DG↗

arXiv2015

Optimal systolic inequalities on Finsler Mobius bands

We prove optimal systolic inequalities on Finsler Mobius bands relating the systole and the height of the Mobius band to its Holmes-Thompson volume. We also establish an optimal systolic in- equality for Finsler Klein bottles of revolution, which we conjecture to hold true for arbitrary Finsler metrics. Extremal metric families both on the Mobius band and the Klein bottle are also presented.

math.DG↗

arXiv2014

Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature

We establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function whose level set volume is bounded in terms of the volume of the manifold. As a consequence of this sweep-out estimate, there exists an embedded, closed (possibly singular) minimal hypersurface whose volume is bounded in terms of the volume of the manifold.

math.DG↗

arXiv2010

Short loop decompositions of surfaces and the geometry of Jacobians

Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions. First, we find bounds on the lengths of homologically independent curves on closed Riemannian surfaces. As a consequence, we show that for any $λ\in (0,1)$ there exists a constant $C_λ$ such that every closed Riemannian surface of genus $g$ whose area is normalized at $4π(g-1)$ has at least $[λg]$ homologically independent loops of length at most $C_λ\log(g)$. This result extends Gromov's asymptotic $\log(g)$ bound on the homological systole of genus $g$ surfaces. We construct hyperbolic surfaces showing that our general result is sharp. We also extend the upper bound obtained by P. Buser and P. Sarnak on the minimal norm of nonzero period lattice vectors of Riemann surfaces %systole of Jacobians of Riemann surfaces in their geometric approach of the Schottky problem to almost $g$ homologically independent vectors. Then, we consider the lengths of pants decompositions on complete Riemannian surfaces in connexion with Bers' constant and its generalizations. In particular, we show that a complete noncompact Riemannian surface of genus $g$ with $n$ ends and area normalized to $4π(g+\frac{n}{2}-1)$ admits a pants decomposition whose total length (sum of the lengths) does not exceed $C_g \, n \log (n+1)$ for some constant $C_g$ depending only on the genus. Finally, we obtain a lower bound on the systolic area of finitely presentable nontrivial groups with no free factor isomorphic to $\Z$ in terms of its first Betti number. The asymptotic behavior of this lower bound is optimal.

math.DG↗

arXiv2008

Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions

We hereby introduce and study the notion of self-contracted curves, which encompasses orbits of gradient systems of convex and quasiconvex functions. Our main result shows that bounded self-contracted planar curves have a finite length. We also give an example of a convex function defined in the plane whose gradient orbits spiral infinitely many times around the unique minimum of the function.

math.DS↗

arXiv2006

Systolic invariants of groups and 2-complexes via Grushko decomposition

We prove a finiteness result for the systolic area of groups, answering a question of M. Gromov. Namely, we show that there are only finitely many possible unfree factors of fundamental groups of~2-complexes whose systolic area is uniformly bounded. Furthermore, we prove a uniform systolic inequality for all 2-complexes with unfree fundamental group that improves the previously known bounds in this dimension.

math.DG↗