The Moser-Trudinger inequality on Kahler-Einstein manifolds
  We prove the conjecture of Tian on the strong form of the Moser-Trudinger inequality for Kahler-Einstein manifolds with positive first Chern class, when there are no holomorphic vector fields, and, more generally, when the setting is invariant under a maximal compact subgroup of the automorphism group. 
D.H. Phong, Jian Song, Jacob Sturm and Ben Weinkove