Related papers: Hyperbolic cusps with convex polyhedral boundary

We prove the existence of equilibrium states for partially hyperbolic endomorphisms with one-dimensional center bundle. We also prove, regarding a class of potentials, the uniqueness of such measures for endomorphisms defined on the 2-torus…

We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding…

In this paper a geometric characterization of the unique geodesics in Thompson's metric spaces is presented. This characterization is used to prove a variety of other geometric results. Firstly, it will be shown that there exists a unique…

We study harmonic maps from a 3-manifold with boundary to $\mathbb{S}^1$ and prove a special case of dihedral rigidity of three dimensional cubes whose dihedral angles are $\pi / 2$. Furthermore we give some applications to mapping torus…

In this paper we prove the conjecture of Alexander and Currier that states, except for covering maps of equidistant surfaces in hyperbolic 3-space, a complete, nonnegatively curved immersed hypersurface in hyperbolic space is necessarily…

We first prove that given a Fuchsian representation $\rho_\circ: \pi_1S \ra {\rm PSL}(2,\R)$, where $S$ is a closed oriented surface of genus $\geq 2$, any hyperbolic cone-metric on $S$ with cone-angles $>2\pi$ isometrically embeds as a…

We prove the existence of $C^{1,1}$ isometric immersions of several classes of metrics on surfaces $(\mathcal{M},g)$ into the three-dimensional Euclidean space $\mathbb{R}^3$, where the metrics $g$ have strictly negative curvature. These…

In this thesis we consider a way to construct a rich family of compact Riemann Surfaces in a combinatorial way. Given a 3-regualr graph with orientation, we construct a finite-area hyperbolic Riemann surface by gluing triangles according to…

We prove that every cusped hyperbolic 3-manifold has a finite cover admitting infinitely many geometric ideal triangulations. Furthermore, every long Dehn filling of one cusp in this cover admits infinitely many geometric ideal…

This paper is the first in a series of two articles whose aim is to extend a recent result of Guillarmou-Lefeuvre on the local rigidity of the marked length spectrum from the case of compact negatively-curved Riemannian manifolds to the…

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 this paper, we prove a local rigidity of convex hypersurfaces in the spaces of constant curvature of dimension $n\ge4$. Namely, we show that two convex isometric hypersurfaces are congruent locally around their corresponding under the…

In this paper, we generalize our results in \cite{GX3} to triangulated surfaces in hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gaussian…

Locally convex compact immersed hypersurfaces in Finsler-Hadamard manifolds with bounded T-curvature are considered. We prove that such hypersurfaces are embedded as the boundary of convex body under certain conditions on the normal…

By a result of W.~P. Thurston, the moduli space of flat metrics on the sphere with $n$ cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension $n-3$. The Hermitian form comes from the area of the…

We prove an existence theorem for convex hypersurfaces of prescribed Gauss curvature in the complement of a compact set in Euclidean space which are close to a cone.

We introduce a new class of discrete conformal structures on surfaces with boundary, which have nice interpolations in 3-dimensional hyperbolic geometry. Then we prove the global rigidity of the new discrete conformal structures using…

Liebmann's Theorem asserts that a compact, connected, convex surface with constant mean curvature (CMC) in the Euclidean space must be a totally umbilical sphere. In this article we extend Liebmann's result to hypersurfaces with boundary.…

The object of study of this article is compact surfaces in the three-dimensional hyperbolic space with a positive-definite second fundamental form. It is shown that several conditions on the Gaussian curvature of the second fundamental form…

We prove the existence of a complete locally Lipschitz continuous hypersurface in weak sense with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under certain assumptions.