Curvature estimates for semi-convex solutions of asymptotic Plateau problem in $\mathbb{H}^{n+1}$

TL;DR

This paper derives curvature estimates for semi-convex hypersurfaces with constant $\sigma_k$ curvature in hyperbolic space, advancing understanding of the asymptotic Plateau problem.

Contribution

It introduces a new concavity inequality for Hessian equations and establishes $C^2$ estimates for solutions with prescribed asymptotic boundary.

Findings

Established $C^2$ estimates for semi-convex hypersurfaces

Derived a new concavity inequality for Hessian equations

Extended results to hypersurfaces with boundary at infinity in hyperbolic space

Abstract

In this paper, we consider the asymptotic $\sigma_k$ Plateau problem in hyperbolic space. We establish $C^2$ estimates for semi-convex complete hypersurfaces satisfying constant $\sigma_k$ curvature with a prescribed asymptotic boundary at the infinity for $2\leq k\leq n-2$ . The result is based on a new crucial concavity inequality derived for hessian equations.