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

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.