Regularity of weak minimizers of the K-energy and applications to properness and K-stability

Abstract

Let (X,ω)(X,ω) be a compact Kahler manifold and HH the space of Kahler metrics cohomologous to ωω. If a csck metric exists in HH, 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 HH 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)(X,L) admitting a csck metric in c1(L)c1(L) is KK-polystable. When the automorphism group is finite, the properness result, combined with a result of Boucksom-Hisamoto-Jonsson, also implies that (X,L)(X,L) is uniformly K-stable.