Constant mean curvature graphs in $\mathbb{H}^3$ defined in exterior domains

TL;DR

This paper proves the existence of constant mean curvature (CMC) graphs in hyperbolic 3-space within exterior domains, extending the understanding of geometric structures with prescribed curvature in hyperbolic geometry.

Contribution

It establishes the existence of CMC $H$ graphs in exterior domains of hyperbolic space, a novel result in the study of geometric analysis in hyperbolic geometry.

Findings

Existence of hyperbolic Killing graphs with prescribed CMC in exterior domains.

Construction of CMC graphs with boundary on an $H$-hypersphere.

Extension of geometric analysis techniques to hyperbolic space.

Abstract

Given $H\in [0,1)$ and given a $C^0$ exterior domain $\Omega$ in a $H-$hypersphere of $\mathbb{H}^3,$ the existence of hyperbolic Killing graphs of CMC $H$ defined in $\overline{\Omega}$ with boundary $ \partial \Omega $ included in the $H-$hypersphere is obtained.