Alexandrov sphere theorems for $ W^{2,n} $-hypersurfaces

Mario Santilli; Paolo Valentini

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arXiv · 2409.01061

Alexandrov sphere theorems for $ W^{2,n} $-hypersurfaces

Mario Santilli·Paolo Valentini

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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.

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BibTeX RIS

Mario Santilli, Paolo Valentini. 2024-09-02. Alexandrov sphere theorems for $ W^{2,n} $-hypersurfaces. https://arxiv.org/abs/2409.01061

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