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. This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.