Preprint A388/2005
Constant mean curvature hypersurfaces in warped product spaces
Luis Alias
We study hypersurfaces with constant mean curvature immersed into warped product spaces of the form $\R\times_\varrho\Pes^n$, where $\Pes^n$ is a complete Riemannian manifold. In particular, our study includes that of constant mean curvature hypersurfaces in product ambient spaces that have been extensively studied recently. It also includes constant mean curvature hypersurfaces in the so called \textit{pseudo-hyperbolic} spaces that resemble hyperbolic space in various senses. If the hypersurface is compact, generalizing previous results by Montiel \cite{mo} we show that the immersion must be a leaf of the trivial totally umbilical foliation $t\in\R\mapsto\{t\}\times\Pes^n$. We also extend a result of Guan and Spruck \cite{gs} from hyperbolic ambient space to the general situation of warped products. This extension allows us to give a slightly more general version of a result by Montiel \cite{mo}, and to derive height estimates for compact constant mean curvature hypersurfaces with boundary in a leaf.