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

Abstract

We show a non existence result for solutions of the prescribed mean curvature equation in the product manifold ${\mathbb{H}}^2\times \mathbb{R}$, 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.

Cited by in corpus (1)

• Constant mean curvature surfaces in ${\mathbb{H}}^2\times \mathbb{R}$ with boundary in two parallel planes