A classical rigidity result of Alexandrov asserts that if $ 1 \leq k \leq n $ is an integer and $ \Sigma $ is a compact $ C^2 $-regular hypersurface of $ \mathbf{R}^{n+1} $ such that the  $ k $-th mean curvature is constant then $ \Sigma $ must be a sphere. In this talk, I discuss an extension of this result to hypersurfaces in $ \mathbf{R}^{n+1} $ which are locally  graphs of $ W^{2,n} $- functions. The proof crucially relies on the theory of currents, in particular on the theory of Legendrian cycles.