On stability and the convergence of the K\"ahler-Ricci flow
  Assuming uniform bounds for the curvature, the exponential convergence of the K\"ahler-Ricci flow is established under two conditions which are a form of stability: the Mabuchi energy is bounded from below, and the dimension of the space of holomorphic vector fields in an orbit of the diffeomorphism group cannot jump up in the limit. 
D.H. Phong and Jacob Sturm