Alexandrov sphere theorems for W2,n-hypersurfaces

In this paper we extend Alexandrov's sphere theorems for higher-order mean curvature functions to W2,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.