REGULARITY OF WEAK MINIMIZERS OF THE K-ENERGY AND APPLICATIONS TO PROPERNESS AND K-STABILITY

Let (X, omega) be a compact Kahler manifold and H the space of Kahler metrics cohomologous to omega. If a csck metric exists in H, we show that all finite energy minimizers of the extended K-energy are smooth csck metrics, partially confirming a conjecture of Y.A. Rubinstein and the second author. As an immediate application, we obtain that the existence of a csck metric in H implies J-properness of the K-energy, thus confirming one direction of a conjecture of Tian. Exploiting this properness result we prove that an ample line bundle (X, L) admitting a csck metric in c(1)(L) is K-polystable. When the automorphism group is finite, the properness result, combined with a result of Boucksom-Hisamoto-Jonsson, also implies that (X, L) is uniformly K-stable.