Interior curvature estimates and the asymptotic plateau problem in hyperbolic space

TL;DR

This paper establishes existence and uniqueness results for convex hypersurfaces in hyperbolic space with prescribed asymptotic boundary and curvature conditions, introducing a new global interior curvature estimate method.

Contribution

It provides a general existence and uniqueness framework for curvature-prescribed hypersurfaces in hyperbolic space, including a novel global interior curvature estimate approach.

Findings

Existence of smooth solutions with bounded hyperbolic principal curvatures.

Uniqueness of solutions when boundary is starshaped or mean convex.

Results extend to De Sitter space via duality.

Abstract

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(\kappa)=\sigma\in(0, 1)$ with a prescribed asymptotic boundary $\Gamma$ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if $\Gamma$ is (Euclidean) starshaped, the solution is unique and also (Euclidean) starshaped while if $\Gamma$ is mean convex the solution is unique. We also show via a strong duality theorem that analogous results hold in De Sitter space. A novel feature of our approach is a "global interior curvature estimate".