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.