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

TL;DR

This paper extends Alexandrov's sphere theorems to $ W^{2,n} $-hypersurfaces with degenerate elliptic conditions, using Legendrian cycles and generalizes umbilicality results for Sobolev hypersurfaces.

Contribution

It introduces a new approach to Alexandrov's theorems for $ W^{2,n} $-hypersurfaces and constructs Legendrian cycles with specific support, answering open questions.

Findings

Extended Alexandrov's sphere theorems to $ W^{2,n} $-hypersurfaces.

Constructed $ n $-dimensional Legendrian cycles with $ 2n $-dimensional support.

Provided a general umbilicality theorem for Sobolev-type hypersurfaces.

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.