Uniqueness of isometric immersions with the same mean curvature
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2018
- DOI:
- 10.48550/arXiv.1802.04244
- arXiv:
- arXiv:1802.04244
- Bibcode:
- 2018arXiv180204244L
- Keywords:
-
- Mathematics - Differential Geometry;
- General Relativity and Quantum Cosmology;
- Mathematics - Analysis of PDEs
- E-Print:
- 23 pages, introduction revised, references updated