Related papers: Manhattan Curves for Hyperbolic Surfaces with Cusp…

We derive basic differential geometric formulae for surfaces in hyperbolic space represented as envelopes of horospheres. The dual notion of parallel hypersurfaces is also studied. The representation is applied to prove existence and…

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 hyperbolic metric for the punctured unit disc in the Euclidean plane is singular at the origin. A renormalization of the metric at the origin is provided by the Euclidean metric. For Riemann surfaces there is a unique germ for the…

A hyperbolic framed curve is a smooth curve with a moving frame in hyperbolic 3-space. It may have singularities. By using this moving frame, we can investigate the differential geometry properties of curves, even at singular points. In…

In this note, we extend the Bridgeman-Kahn identity to all finite-volume orientable hyperbolic $n$-manifolds with totally geodesic boundary. In the compact case, Bridgeman and Kahn are able to express the manifold's volume as the sum of a…

We prove an extension of Basmajian's identity to $n$-Hitchin representations of compact bordered surfaces. For $n=3$, we show that this identity has a geometric interpretation for convex real projective structures analogous to Basmajian's…

We introduce a notion of relative isospectrality for surfaces with boundary having possibly non-compact ends either conformally compact or asymptotic to cusps. We obtain a compactness result for such families via a conformal surgery that…

We get new results (and rederive some know ones) on smooth surfaces in $\mathbb{R}^n$ by unifying several view points into a coherent general view. Namely, we show and use new relations of the evolute (caustic) with the curvature ellipse,…

In this survey paper we give a proof of hyperbolicity of the complex of curves for a non-exceptional surface S of finite type combining ideas of Masur/Minsky and Bowditch. We also shortly discuss the relation between the geometry of the…

We examine the large systole problem, which concerns compact hyperbolic Riemannian surfaces whose systole, the length of the shortest noncontractible loops, grows logarithmically in genus. The generalization of a construction of Buser and…

In machine learning, data is usually represented in a (flat) Euclidean space where distances between points are along straight lines. Researchers have recently considered more exotic (non-Euclidean) Riemannian manifolds such as hyperbolic…

This article is the second in a series of two whose aim is to extend a recent result of Guillarmou-Lefeuvre [arXiv:1806.04218] on the local rigidity of the marked length spectrum from the case of compact negatively-curved Riemannian…

We show that for every non-elementary hyperbolic group, an associated topological flow space admits a coding based on a transitive subshift of finite type. Applications include regularity results for Manhattan curves, the uniqueness of…

We make observations about constant mean curvature surfaces in Euclidean 3-space and their dual surfaces, and the resulting pairs of surfaces in hyperbolic 3-space under the Lawson correspondence.

We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples include the hyperbolic plane, the hyperbolic disk,…

For a Riemannian manifold $M$, possibly with boundary, we consider the Riemannian product $M\times\mathbb{R}^k$ with a smooth positive function that weights the Riemannian measures. In this work we characterize parabolic hypersurfaces with…

We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed…

We show that the index of a constant mean curvature 1 surface in hyperbolic 3-space is completely determined by the compact Riemann surface and secondary Gauss map that represent it in Bryant's Weierstrass representation. We give three…

We derive a sharp cusp count for finite volume complex hyperbolic surfaces which admit smooth toroidal compactifications. We use this result, and the techniques developed in [DiC12], to study the geometry of cusped complex hyperbolic…

We show that Norbury's McShane identity for nonorientable cusped hyperbolic surfaces N generalizes to quasifuchsian representations of pi_1(N) as well as pseudo-Anosov mapping Klein bottles with singular fibers given by N.