Hypersurfaces of constant sum Hessian curvature in Hyperbolic space

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.