Height estimates for H-surfaces in the warped product $${\mathbb {M}}\times _{f} {\mathbb {R}}$$ M × f 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.