Preprint A388/2005
Constant mean curvature hypersurfaces in warped product spaces

Luis Alias

**Keywords: **

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.