Height estimates for $H$-surfaces in the warped product $\mathbb{M}{\times }_f\mathbb{R}$

Abstract

In this article, we consider compact surfaces $\Sigma$ having constant mean curvature $H$ ($H$-surfaces) whose boundary $\Gamma =\partial \Sigma \subset {\mathbb{M}}_0=\mathbb{M}{\times }_f\{0\}$ is transversal to the slice ${\mathbb{M}}_0$ of the warped product $\mathbb{M}{\times }_f\mathbb{R}$, here $\mathbb{M}$ denotes a Hadamard surface. We obtain height estimate for a such surface $\Sigma$ having positive constant mean curvature involving the area of a part of $\Sigma$ above of ${\mathbb{M}}_0$ and the volume it bounds. Also we give general conditions for the existence of rotationally-invariant topological spheres having positive constant mean curvature $H$ in the warped product $\mathbb{H}{\times }_f\mathbb{R}$, where $\mathbb{H}$ denotes the hyperbolic disc. Finally we present a non-trivial example of such spheres.