Hypersurfaces of constant sum Hessian curvature in Hyperbolic space

TL;DR

This paper investigates the existence and properties of complete hypersurfaces in hyperbolic space with constant sum Hessian curvature, solving a geometric boundary value problem related to the asymptotic Plateau problem.

Contribution

It introduces a new class of hypersurfaces characterized by a specific curvature condition involving the sum of Hessian eigenvalues in hyperbolic space.

Findings

Existence of hypersurfaces with prescribed boundary and curvature conditions.

Characterization of asymptotic behavior of these hypersurfaces.

Extension of classical Plateau problem to Hessian curvature settings.

Abstract

In this paper, we study the asymptotic Plateau problem in hyperbolic space for constant sum Hessian curvature. More precisely, given a asymptotic boundary $\Gamma$, one seeks a complete hypersurface $\Sigma$ in $\mathbb{H}^{n+1}$ satisfying $\sigma_{n-1}(\kappa)+\alpha\sigma_{n}(\kappa)=\sigma\in (0,n),\,\,\partial \Sigma=\Gamma$ where $\alpha$ is a non-negative number.