Rigidity of volume-minimising hypersurfaces in Riemannian 5-manifolds
Abstract
In this paper we generalize the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimizing closed hypersurface $\Sigma$ of a Riemannian 5-manifold $M$ with scalar curvature bounded from below by a positive constant in terms of the total traceless Ricci curvature of $\Sigma$. Furthermore, if $\Sigma$ saturates the respective upper bound and $M$ has nonnegative Ricci curvature, then $\Sigma$ is isometric to $\mathbb{S}^4$ up to scaling and $M$ splits in a neighborhood of $\Sigma$. Also, we obtain a rigidity result for the Riemannian cover of $M$ when $\Sigma$ minimizes the volume in its homotopy class and saturates the upper bound.
- Publication:
-
Mathematical Proceedings of the Cambridge Philosophical Society
- Pub Date:
- September 2019
- DOI:
- 10.1017/S0305004118000361
- arXiv:
- arXiv:1703.00930
- Bibcode:
- 2019MPCPS.167..345M
- Keywords:
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- Mathematics - Differential Geometry
- E-Print:
- 9 pages. Minor changes. Version to appear in Math. Proc. Cambridge Philos. Society