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

Abstract We prove that the proximal unit normal bundle of the subgraph of a $W^{2,n} $ -function carries a natural structure of Legendrian cycle . This result is used to obtain an Alexandrov-type sphere theorem for hypersurfaces in $ \mathbf{R}^{n+1} $ , which are locally graphs of arbitrary $W^{2,n} $ -functions. We also extend the classical umbilicality theorem to $ W^{2,1} $ -graphs, under the Lusin (N) condition for the graph map.