Related papers: Flat surfaces in hyperbolic 3-space whose hyperbol…

By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic $3$-manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic…

Let $M$ be a closed hyperbolic 3-manifold that admits no infinitesimal conformally-flat deformations. Examples of such manifolds were constructed by Kapovich. Then if $g$ is a Riemannian metric on $M$ with scalar curvature greater than or…

On a closed Riemannian surface of negative curvature, we prove a characterization for configurations of closed geodesics arising from one parameter Allen-Cahn min-max constructions. One of the facts we conclude is that every geodesic occurs…

A sequence of distinct closed surfaces in a hyperbolic 3-manifold M is asymptotically geodesic if their principal curvatures tend uniformly to zero. When M has finite volume, we show such sequences are always asymptotically dense in the…

We study minimal Lagrangian surfaces in the complex hyperbolic quadric. We show that minimality of a Lagrangian surface is characterized by a loop of flat connections, which yields an associated $\mathbb S^1$-family of isometric…

In this work, we are interested in the differential geometry of surfaces in simply isotropic $\mathbb{I}^3$ and pseudo-isotropic $\mathbb{I}_{\mathrm{p}}^3$ spaces, which consists of the study of $\mathbb{R}^3$ equipped with a degenerate…

It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a "geometric" triangulation of the manifold). Under a mild homology assumption on the manifold we construct…

In this paper, we consider the contracting curvature flow of smooth closed surfaces in $3$-dimensional hyperbolic space and in $3$-dimensional sphere. In the hyperbolic case, we show that if the initial surface $M_0$ has positive scalar…

We define and prove the existence of unique solutions of an asymptotic Plateau problem for spacelike maximal surfaces in the pseudo-hyperbolic space of signature (2, n): the boundary data is given by loops on the boundary at infinity of the…

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the…

Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian 3-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These "minimizing"…

We consider geometric triangulations of surfaces, i.e., triangulations whose edges can be realized by disjoint locally geodesic segments. We prove that the flip graph of geometric triangulations with fixed vertices of a flat torus or a…

This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $\varepsilon$-fills the surface.

Let M be a closed hyperbolic three manifold. We construct closed surfaces which map by immersions into M so that for each one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding…

In this article we consider surfaces in the product space $\h^2\times \r$ of the hyperbolic plane $\h^2$ with the real line. The main results are: a description of some geometric properties of minimal graphs; new examples of complete…

We prove rigidity results involving the Hawking mass for surfaces immersed in a $3$-dimensional, complete Riemannian manifold $(M,g)$ with non-negative scalar curvature (resp. with scalar curvature bounded below by $-6$). Roughly, the main…

On a finite-volume hyperbolic $3$-manifold, we establish an upper bound on the area of closed embedded surfaces with constant mean curvature at least one, depending on the mean curvature and the genus bounds. This area bound implies…

Let M be a hyperbolic 3-manifold with nonempty totally geodesic boundary. We prove that there are upper and lower bounds on the diameter of the skinning map of M that depend only on the volume of the hyperbolic structure with totally…

We review recent results on classifying complete constant mean curvature 1 (CMC 1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature" -- one is the total absolute curvature, which is…

Let N be a topologically finite, orientable 3-manifold with ideal triangulation. We show that if there is a solution to the hyperbolic gluing equations, then all edges in the triangulation are essential. This result is extended to a…