arXiv subjects

Nicolas Tholozan

Publications and source records attributed to Nicolas Tholozan.

21 records · Page 2Linked to original sources

arXiv2015

Entropy of Hilbert metrics and length spectrum of Hitchin representations in $\mathrm{PSL}(3,\mathbb{R})$

We prove a sharp inequality between the Blaschke and Hilbert distance on a proper convex domain: for any two points $x$ and $y$, \[d^B(x,y) < d^H(x,y) +1.\] We obtain two interesting consequences: the first one is the volume entropy rigidity for Hilbert geometries : for any proper convex domain of $\mathbb{R}\mathbf{P}^n$, the volume of a ball of radius $R$ grows at most like $e^{(n-1)R}$. The second consequence is the following fact: for any Hitchin representation of a surface group into $\mathrm{PSL}(3,\mathbb{R})$, there exists a Fuchsian representation $j$ in $\mathrm{PSL}(2,\mathbb{R})$ such that the length spectrum of $j$ is uniformly smaller than the length spectrum of $ρ$.

math.MG↗

arXiv2014

Dominating surface group representations and deforming closed AdS 3-manifolds

In a previous paper by Deroin-Tholozan, the authors construct a map $\mathbfΨ_ρ$ from the Teichmüller space of $S$ to itself and prove that, when $M$ has sectional curvature $\leq -1$, the image of $\mathbfΨ_ρ$ lies (almost always) in the domain $\mathrm{Dom}(ρ)$ of Fuchsian representations stricly dominating $ρ$. Here we prove that $\mathbfΨ_ρ: \mathrm{Teich}(S) \to \mathrm{Dom}(ρ)$ is a homeomorphism. As a consequence, we are able to describe the topology of the deformation space of anti-de Sitter structures on closed 3-manifolds.

math.GT↗

arXiv2013

Sur la complétude de certaines variétés pseudo-riemanniennes localement symétriques

We prove that some symetric semi-riemannian manifolds do not admit a proper domain which is divisible by the action of a discrete group of isometries. In other words, if a closed semi-riemannian manifold is locally isometric to such a model, and if its developing map is injective, then the manifold is actually geodesically complete, and therefore isometric to a quotient of the whole model space. Those results extend, under additional hypothesis (the injectivity of the developing map), the theorems of Carriere and Klingler stating that closed lorentz manifolds of constant curvature are geodesically complete.

math.DG↗