Instability results for an elliptic equation on compact Riemannian   manifolds with non-negative Ricci curvature
  We prove nonexistence of nonconstant local minimizers for a class of functionals, which typically appears in the scalar two-phase field model, over a smooth N-dimensional Riemannian manifold without boundary with non-negative Ricci curvature. Conversely for a class of surfaces possessing a simple closed geodesic along which the Gauss curvature is negative we prove existence of nonconstant local minimizers for the same class of functionals. 
Arnaldo Nascimento, Alexandre Gon\c{c}alves