Constant mean curvature surfaces in warped product manifolds

TL;DR

This paper proves that constant mean curvature surfaces in specific warped product manifolds are umbilic, extending classical results and applying to important spacetime models like deSitter-Schwarzschild.

Contribution

It generalizes the Alexandrov theorem to warped product manifolds, identifying conditions under which CMC surfaces are necessarily umbilic.

Findings

Any CMC surface in the considered manifolds is umbilic under certain conditions

Results apply to deSitter-Schwarzschild and Reissner-Nordstrom manifolds

Extends classical Euclidean results to more complex geometries

Abstract

We consider surfaces with constant mean curvature in certain warped product manifolds. We show that any such surface is umbilic, provided that the warping factor satisfies certain structure conditions. This theorem can be viewed as a generalization of the classical Alexandrov theorem in Euclidean space. In particular, our results apply to the deSitter-Schwarzschild and Reissner-Nordstrom manifolds.