Related papers: Adapted Metrics for Codimension one Singular Hyper…

We prove that if $f:\mathbb{B}^n \to \mathbb{B}^n$, for $n\geq 2$, is a homeomorphism with bounded skew over all equilateral hyperbolic triangles, then $f$ is in fact quasiconformal. Conversely, we show that if $f:\mathbb{B}^n \to…

We study the Bernoulli property for a class of partially hyperbolic systems arising from skew products. More precisely, we consider a hyperbolic map $(T,M,\mu)$, where $\mu$ is a Gibbs measure, an aperiodic H\"older continuous cocycle…

We investigate the rigidity of hyperbolic cone metrics on $3$-manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics.…

We obtain a exponential large deviation upper bound for continuous observables on suspension semiflows over a non-uniformly expanding base transformation with non-flat singularities and/or discontinuities, where the roof function defining…

We consider constrained partial differential equations of hyperbolic type with a small parameter $\varepsilon>0$, which turn parabolic in the limit case, i.e., for $\varepsilon=0$. The well-posedness of the resulting systems is discussed…

We establish the existence and finiteness of equilibrium states for a class of partially hyperbolic endomorphisms. In our first result, we assume that the central direction is simple. In the second result, we consider the case where there…

We derive geometric formulas for the mass of asymptotically hyperbolic manifolds using coordinate horospheres. As an application, we obtain a new rigidity result of hyperbolic space: if a complete asymptotically hyperbolic manifold has…

On finite-volume hyperbolic $3$-manifolds, we compare volumes of different metrics using the exponential convergence of Ricci-DeTurck flow toward the hyperbolic metric $h_0$. We prove that among metrics with scalar curvature bounded below…

For any geodesic metric space $X$, we give a complete cohomological characterisation of the hyperbolicity of $X$ in terms of vanishing of its second $\ell^{\infty}$-cohomology. We extend this result to the relative setting of $X$ with a…

A {\em singular hyperbolic set} is a partially hyperbolic set with singularities (all hyperbolic) and volume expanding central direction \cite{MPP1}. We study connected, singular-hyperbolic, attracting sets with dense closed orbits {\em and…

In this paper, we study physical measures for partially hyperbolic diffeomorphisms with multi one-dimensional centers under the condition that all Gibbs $u$-states are hyperbolic. We prove the finiteness of ergodic physical measures. Then…

The statistical properties of mostly expanding partially hyperbolic diffeomorphisms have been substantially studied. In this paper, we would like to address the entropy properties of mostly expanding partially hyperbolic diffeomorphisms. We…

We present a new criterion for the complex hyperbolicity of a non-compact quotient X of a bounded symmetric domain. For each p $\ge$ 1, this criterion gives a precise condition under which the subvarieties V $\subset$ X with dim V $\ge$ p…

Let $C$ be an $m$-dimensional cone immersed in $\mathbb{R}^{n+m}$. In this paper, we show that if $F:M^m \rightarrow \mathbb{R}^{n+m}$ is a properly immersed mean curvature flow self-shrinker which is smoothly asymptotic to $C$, then it is…

This paper establishes the existence of a gap for the stable length spectrum on a hyperbolic manifold. If M is a hyperbolic n-manifold, for every positive e there is a positive d depending only on n and on e such that an element of pi_1(M)…

Stable accessibility for partially hyperbolic diffeomorphisms is central to their ergodic theory, and we establish its \(C^1\)-density among 1. all, 2. volume-preserving, 3. symplectic, and 4. contact partially hyperbolic flows. As…

Let $S\subset \C^n$, $n\geq 3$ be a compact connected 2-codimensional submanifold having the following property: there exists a Levi-flat hypersurface whose boundary is $S$, possibly as a current. Our goal is to get examples of such $S$…

The precise behavior of the quasi-hyperbolic metric near a $\mathcal C^{1,1}$-smooth part of the boundary of a domain in $\mathbb{R}^n$ is obtained.

Suppose n>2, let M,M' be n-dimensional connected complete finite-volume hyperbolic manifolds with non-empty geodesic boundary, and suppose that the fundamental group of M is quasi-isometric to the fundamental group of M' (with respect to…

Recently, the explicit volume formulae for hyperbolic cone-manifolds, whose underlying space is the 3-sphere and the singular set is the knot $4_1$ and the links $5^2_1$ and $6^2_2$, have been obtained by the second named author and his…