Entire spacelike constant $\sigma_k$ curvature hypersurfaces with prescribed boundary data at infinity

TL;DR

This paper establishes existence and uniqueness results for convex, entire, spacelike hypersurfaces with constant _k curvature in hyperbolic space, prescribed boundary data at infinity, and perturbations, extending known results to new cases.

Contribution

It proves the existence of such hypersurfaces with prescribed boundary directions and perturbations, including the case of constant Gauss curvature, and establishes uniqueness under certain conditions.

Findings

Existence of hypersurfaces with prescribed boundary at infinity.

Extension of results to constant Gauss curvature case.

Uniqueness of solutions under bounded perturbations.

Abstract

In this paper, we investigate the existence and uniqueness of convex, entire, spacelike hypersurfaces of constant $\sigma_k$ curvature with prescribed set of lightlike directions $\mathcal{F}\subset\mathbb{S}^{n-1}$ and perturbation $q$ on $\mathcal{F}$. We prove that given a closed set $\mathcal{F}$ in the ideal boundary at infinity of hyperbolic space and a perturbation $q$ that satisfies some mild conditions, there exists a complete entire spacelike constant $\sigma_k$ curvature hypersurface $\mathcal{M}_u$ with prescribed set of lightlike directions $\mathcal{F}$ satisfying when $\frac{x}{|x|}\in\mathcal{F},$ as $|x|\rightarrow\infty,$ $u(x)-|x|\rightarrow q\left(\frac{x}{|x|}\right).$ This result is new even for the case of constant Gauss curvature. We also prove that when the Gauss map image is a half disc $\bar{B}_1^+$ and the perturbation $q\equiv 0,$ if a CMC hypersurface $\mathcal{M}_u$ satisfies $|u(x)-V_{\bar{\mathcal{B}}_+}(x)|$ is bounded, then $u(x)$ is unique.