Legendrian cycles and Alexandrov sphere theorems for W^{2,n}-hypersurfaces

In this thesis, we prove that the proximal unit normal bundle of the graph of a W^{2,n}-function in n-variables carries a natural structure of Legendrian cycle. We then generalize Alexandrov's sphere theorems for higher-order mean curvature functions to hypersurfaces in \mathbb{R}^{n+1} which are locally graphs of arbitrary W^{2,n}-functions, under a general degenerate ellipticity condition. The proof relies on extending the Montiel-Ros argument to this class of hypersurfaces and on the existence of the aforementioned Legendrian cycles. We also prove the existence of n-dimensional Legendrian cycles with 2n-dimensional support, thus answering a question posed by Rataj and Zähle. Furthermore, we extend some of these results to Sobolev-type manifolds, representable as finite unions of W^{2,n}-regular graphs, and generalize Reilly's variational formulas in this context. Finally, we provide a very general version of the umbilicality theorem for Sobolev-type hypersurfaces.