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 $\mathbb{H}^2 \times \R$, 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.