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 ${\mathbb {M}}\times _f{\mathbb {R}}$, where ${\mathbb {M}}$ 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.