A remark on constant mean curvature hypersurfaces in warped product manifolds

TL;DR

This paper extends Alexandrov's theorem to warped product manifolds, proving a rigidity result that characterizes constant mean curvature hypersurfaces, and offers an alternative proof to a recent related theorem.

Contribution

It generalizes Alexandrov's theorem to warped product manifolds and provides a new proof of a recent related result by Brendle.

Findings

Proves a rigidity result for constant mean curvature hypersurfaces in warped products.

Generalizes previous proofs by Reilly and Ros.

Offers an alternative proof to Brendle's recent theorem.

Abstract

Alexandrov's theorem asserts that spheres are the only closed embedded constant mean curvature hypersurfaces in space forms. In this paper, we consider Alexandrov's theorem in warped product manifolds and prove a rigidity result in the spirit of Alexandrov's theorem. Our approach generalizes the proofs of Reilly and Ros and, under more restrictive assumptions, it provides an alternative proof of a recent theorem of Brendle.