Uniqueness of isometric immersions with the same mean curvature

TL;DR

This paper investigates the rigidity of isometric immersions with identical mean curvature into warped product spaces, showing that such hypersurfaces are uniquely determined up to rotation in certain gravitational models.

Contribution

It establishes new rigidity results for hypersurfaces with the same mean curvature in warped product spaces, extending to $\sigma_2$-curvature conditions.

Findings

Hypersurfaces with the same mean curvature are rotationally symmetric in Schwarzschild spaces.

Uniqueness results hold for star-shaped hypersurfaces with nonzero mass.

Similar rigidity applies under $\sigma_2$-curvature conditions.

Abstract

Motivated by the quasi-local mass problem in general relativity, we study the rigidity of isometric immersions with the same mean curvature into a warped product space. As a corollary of our main result, two star-shaped hypersurfaces in a spatial Schwarzschild or AdS-Schwarzschild manifold with nonzero mass differ only by a rotation if they are isometric and have the same mean curvature. We also give similar results if the mean curvature condition is replaced by an $\sigma_2$-curvature condition.