Smooth Solutions to Asymptotic Plateau Type Problem in Hyperbolic Space

TL;DR

This paper proves the existence of smooth hypersurfaces with prescribed Weingarten curvature in hyperbolic space, extending the asymptotic Plateau problem to cases with non-constant curvature and specific boundary conditions.

Contribution

It provides new existence results for hypersurfaces with prescribed curvature in hyperbolic space under certain boundary and convexity conditions, generalizing previous work.

Findings

Existence of solutions for $k=n$ with convex boundary

Existence of solutions for $k<n$ with disk boundary

Utilization of Pogorelov interior second order estimates

Abstract

We investigate on the existence of smooth complete hypersurface with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under the assumption that there exists an asymptotic subsolution. We give an affirmative answer for the case $k = n$ when the asymptotic boundary $\Gamma$ bounds a uniformly convex domain, and for $k < n$ when $\Gamma$ bounds a disk, utilizing Pogorelov type interior second order estimate. Our result complements our previous work \cite{Sui2019, Sui-Sun}, and generalizes the asymptotic Plateau type problem to non-constant prescribed curvature case.