1. Alexandrov sphere theorems for $ W^{2,n} $-hypersurfaces
- Author
-
Santilli, Mario and Valentini, Paolo
- Subjects
Mathematics - Differential Geometry ,53C24, 53C65, 49Q20 - Abstract
In this paper we extend Alexandrov's sphere theorems for higher-order mean curvature functions to $ W^{2,n} $-regular hypersurfaces under a general degenerate elliptic condition. The proof is based on the extension of the Montiel-Ros argument to the aforementioned class of hypersurfaces and on the existence of suitable Legendrian cycles over them. Using the latter we can also prove that there are $ n $-dimensional Legendrian cycles with $ 2n $-dimensional support, hence answering a question by Rataj and Zaehle. Finally we provide a very general version of the umbilicality theorem for Sobolev-type hypersurfaces., Comment: New introduction. Acknowledgments added
- Published
- 2024