Constant mean curvature graphs on exterior domains of the hyperbolic plane

TL;DR

This paper establishes existence results for non-rotational constant mean curvature surfaces in hyperbolic plane products, using advanced PDE techniques and analyzing asymptotic behaviors of these surfaces.

Contribution

It provides new existence theorems for constant mean curvature ends in hyperbolic space and investigates their asymptotic properties, extending previous work on rotational ends.

Findings

Existence of non-rotational constant mean curvature ends in        imes  

Application of Schauder theory and continuity method to exterior domains in hyperbolic space

Analysis of asymptotic behavior of rotational ends by Sa Earp and Toubiana

Abstract

We prove an existence result for non rotational constant mean curvature ends in $\mathbb{H}^2 \times \mathbb{R}$, where $\mathbb{H}^2$ is the hyperbolic real plane. The value of the curvature is $h \, \in \, (0, 1/2)$. We use Schauder theory and a continuity method for solution of the prescribed mean curvature equation of exterior domains of $\mathbb{H}^2$. We also prove a fine property of the asymptotic behavior of the rotational ends introduced by Sa Earp and Toubiana.