Rigidity for critical points in the Levy-Gromov inequality
  The Levy-Gromov inequality states that round spheres have the least isoperimetric profile (normalized by total volume) among Riemannian manifolds with a fixed positive lower bound on the Ricci tensor. In this note we study critical metrics corresponding to the Levy-Gromov inequality and prove that, in two-dimensions, this criticality condition is quite rigid, as it characterizes round spheres and projective planes. 
Fabio Cavalletti, Francesco Maggi, Andrea Mondino
math.DG math.OC