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Gilles Courtois

Publications and source records attributed to Gilles Courtois.

14 recordsLinked to original sources
arXiv2026

Quantizing Geodesics in Kähler and Sasaki Geometry

The space of Kähler potentials can be quantized through the classical Fubini-Study map, relating infinite-dimensional geometric structures to finite-dimensional symmetric spaces. We prove (exactly) when the Fubini-Study image of a geodesic line in the space of positive definite Hermitian matrices gives rise to a quasi-geodesic in the space of Kähler potentials. Furthermore, we introduce a quantization procedure for geodesics between potentials on normal Kähler varieties and show how this construction extends to the Sasaki setting.

math.DG↗
arXiv2025

Volume growth of horospheres in diagonalizable Heintze groups

We study the volume growth of horospheres in a Heintze group of the form R ___ A R d with A a diagonal derivation. We conclude that the isometry and quasi-isometry classes of horospheres (with their intrinsic geometry) coincide. Furthermore, if A is not a scalar multiple of the identity, then there are exactly two such classes, characterized by their volume growth, which we calculate explicitly.

math.DG↗
arXiv2024

On Horospherical Rigidity

We provide intrinsic conditions on the geometry of horospheres in a closed, negatively curved Riemannian manifold of dimension greater than or equal to 3, which guarantee that the sectional curvature is constant.

math.DG↗
arXiv2024

Hausdorff dimension, diverging Schottky representations and the infinite dimensional hyperbolic space

One of our main goals in this paper is to understand the behavior of limit sets of a diverging sequence of Schottky groups in the group of isometries of the N-dimensional hyperbolic space. This leads us to a generalization of a classical theorem of Bowen on variations of Hausdorff dimension of limit sets; and to a method of transforming a diverging sequence of Schottky groups into an almost converging sequence in the group of isometries of the infinite dimensional hyperbolic space. Our results apply in particular to an example of McMullen and generalize a previous work by Mehmeti and Dang.

math.DS↗
arXiv2024

Rigidity of flat holonomies

We prove that the existence of one horosphere in the universal cover of a closed, strictly quarter pinched, negatively curved Riemannian manifold of dimension $n\geq 3$ on which the stable holonomy along minimizing geodesics coincide with the Riemannian parallel transport, implies that the manifold is homothetic to a real hyperbolic manifold.

math.DG↗
arXiv2021

Finiteness Theorems for Gromov-Hyperbolic Spaces and Groups

In this article we prove that the set of torsion-free groups acting by isometries on a hyperbolic metric space whose entropy is bounded above and with a compact quotient is finite. The number of such groups can be estimated in terms of the hyperbolicity constant and of an upper bound of the entropy of the space and of an upper bound of the diameter of its quotient. As a consequence we show that the set of non cyclic torsion-free $δ$-hyperbolic marked groups whose entropy is bounded above by a number $H$ is finite with cardinality depending on $δ$ and $H$ alone. From these results, we draw homotopical and topological finiteness theorems for compact metric spaces and manifolds.

math.GR↗
arXiv2020

Curvature-Free Margulis Lemma for Gromov-Hyperbolic Spaces

We prove curvature-free versions of the celebrated Margulis Lemma. We are interested by both the algebraic aspects and the geometric ones, with however an emphasis on the second and we aim at giving quantitative (computable) estimates of some important invariants. Our goal is to get rid of the pointwise curvature assumptions in order to extend the results to more general spaces such as certain metric spaces. Essentially the upper bound on the curvature is replaced by the assumption that the space is hyperbolic in the sense of Gromov and the lower bound of the curvature by an upper bound on the entropy which we recall the definition.

math.DG↗
arXiv2017

Isospectral finiteness on convex cocompact hyperbolic 3-manifolds

In this paper we show that a given set of lengths of closed geodesics, there are only finitely many convex cocompact hyperbolic 3-manifolds with that specified length spectrum, homotopy equivalent to a given 3-manifold without a handlebody factor, up to orientation preserving isometries.

math.GT↗
arXiv2010

Differentiable Rigidity under Ricci curvature lower bound

In this article we prove a differentiable rigidity result. Let $(Y, g)$ and $(X, g_0)$ be two closed $n$-dimensional Riemannian manifolds ($n\geqslant 3$) and $f:Y\to X$ be a continuous map of degree $1$. We furthermore assume that the metric $g_0$ is real hyperbolic and denote by $d$ the diameter of $(X,g_0)$. We show that there exists a number $\varepsilon:=\varepsilon (n, d)>0$ such that if the Ricci curvature of the metric $g$ is bounded below by $-n(n-1)$ and its volume satisfies $\vol_g (Y)\leqslant (1+\varepsilon) \vol_{g_0} (X)$ then the manifolds are diffeomorphic. The proof relies on Cheeger-Colding's theory of limits of Riemannian manifolds under lower Ricci curvature bound.

math.DG↗
arXiv2008

Uniform growth of groups acting on Cartan-Hadamard spaces

Let $X$ be an $n$-dimensional simply connected manifold of pinched sectional curvature $-a^2 \leq K \leq -1$. There exist a positive constant $C(n,a)$ such that for any finitely generated discrete group $Γ$ acting on $X$, then either $Γ$ is virtually nilpotent or the algebraic entropy $Ent (Γ) \geq C(n,a)$.

math.DG↗
arXiv2005

Rigidity of amalgamated product in negative curvature

Let $Γ$ be the fundamental group of a compact n-dimensional riemannian manifold X of sectional curvature bounded above by -1. We suppose that $Γ$ is a free product of its subgroup A and B over the amalgamated subgroup C. We prove that the critical exponent $δ(C)$ of C satisfies $δ(C) \geq n-2$. The equality happens if and only if there exist an embedded compact hypersurface Y in X, totally geodesic, of constant sectional curvature -1, with fundamental group C and which separates X in two connected components whose fundamental groups are A and B. Similar results hold if $Γ$ is an HNN extension, or more generally if $Γ$ acts on a simplicial tree without fixed point.

math.DG↗