Non existence of constant mean curvature graphs on circular annuli of $\mathbb{H}^2$

TL;DR

This paper proves non-existence results for constant mean curvature graphs on circular annuli in hyperbolic space, providing a priori estimates that depend on geometric parameters and mean curvature values.

Contribution

It establishes new a-priori estimates and non-existence results for solutions of the prescribed mean curvature equation in hyperbolic space, specifically on annuli.

Findings

No solutions for certain mean curvature values on annuli

A-priori bounds depend on annulus thickness and boundary data

Results cover all mean curvature values in (0, 1/2]

Abstract

We show a non existence result for solutions of the prescribed mean curvature equation in the product manifold $\mathbb{H}^2 \times \R$, where $\mathbb{H}^2$ is the real hyperbolic plane. More precisely we prove a-priori estimates for graphs with constant mean curvature $h \in (0, 1/2]$ on circular annuli of $\mathbb{H}^2$. For $0 < h < 1/2$ we obtain an estimate from above on any circular annulus and one from below on annuli with a small hole, the size of the hole depending on $h$. For $h = 1/2$ we obtain both estimates for any circular annulus. All the estimates depend only on the thickness of the annulus and the value of the graph on the outer boundary.