Stable solutions of elliptic equations on Riemannian manifolds
  This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new weighted Poincar\'e inequality which allows to prove Liouville type results and the flatness of the level sets of the solution in dimension 2, under suitable geometric assumptions on the ambient manifold. 
Alberto Farina, Yannick Sire and Enrico Valdinoci