Related papers: Asymmetric expansion preserves hyperbolic convexit…

In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known…

We show that a relatively hyperbolic group either is virtually cyclic or has uniform exponential growth.

We consider curvature flows in hyperbolic space with a monotone, symmetric, homogeneous of degree 1 curvature function F. Furthermore we assume F to be either concave and inverse concave or convex. For compact initial hypersurfaces, which…

We study the supremum of the volume of hyperbolic polyhedra with some fixed combinatorics and with vertices of any kind (real, ideal or hyperideal). We find that the supremum is always equal to the volume of the rectification of the…

Hyperbolic polynomials are real polynomials whose real hypersurfaces are nested ovaloids, the inner most of which is convex. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential…

We review recent progress on two closely related sets of questions concerning convex co-compact hyperbolic manifolds, or convex domains in those manifolds, such as their convex core. The first set of questions is to what extent the…

Asymptotic expansions as well as necessary and sufficient conditions are provided for the pointwise convergence of the spherical partial integrals of the associated Fourier transforms on the real hyperbolic space. The proposed method…

Special relativity corresponds to hyperbolic geometry at constant velocity while the so-called general relativity corresponds to hyperbolic geometry of uniformly accelerated systems. Generalized expressions for angular momentum, centrifugal…

A family of orthogonal polynomials on the disk (which we call scattering polynomials) serves to formulate a remarkable Fourier expansion of the composition of a sequence of Poincar\'e disk automorphisms. Scattering polynomials are tied to…

A representation of a finitely generated group into the projective general linear group is called convex co-compact if it has finite kernel and its image acts convex co-compactly on a properly convex domain in real projective space. We…

The purpose of the present paper is to show few examples of nonlinear PDEs (mostly with strong geometric features) for which there is a hidden convex structure. This is not only a matter of curiosity. Once the convex structure is…

The convex body isoperimetric conjecture in the plane asserts that the least perimeter to enclose given area inside a unit disk is greater than inside any other convex set of area $\pi$. In this note we confirm two cases of the conjecture:…

We study the convergence of an axially symmetric hypersurface evolving by volume preserving mean curvature flow. Assuming the surface is not pinching off along the axis at any time during the flow, and without any additional conditions, as…

We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued…

In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of…

In hyperbolic dynamics, a well-known result is: every hyperbolic Lyapunov stable set, is attracting; it's natural to wonder if this result is maintained in the sectional-hyperbolic dynamics. This question is still open, although some…

The Stoker problem, first formulated in 1968, consists in understanding to what extent a convex polyhedron is determined by its dihedral angles. By means of the double construction, this problem is intimately related to rigidity issues for…

This paper explores the asymptotic behaviour of the radii of convexity and uniform convexity for normalized Bessel functions with respect to large order. We provide detailed asymptotic expansions for these radii and establish recurrence…

We provide a general construction of convex cocompact hyperbolic reflection groups with three-dimensional limit sets. More precisely, our construction takes as input an arbitrary simplicial complex L of dimension 3 on n vertices, and…

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the…