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

TL;DR

This paper proves that all finite energy minimizers of the extended K-energy are smooth constant scalar curvature Kähler (cscK) metrics on compact Kähler manifolds, confirming parts of conjectures relating to properness and stability.

Contribution

It demonstrates that finite energy minimizers of the extended K-energy are smooth cscK metrics, and establishes the link between existence of cscK metrics, properness of the K-energy, and K-stability.

Findings

Finite energy minimizers are smooth cscK metrics.

Existence of cscK implies J-properness of the K-energy.

Ample line bundles with cscK metrics are K-polystable.

Abstract

Let $(X,\omega)$ be a compact K\"ahler manifold and $\mathcal H$ the space of K\"ahler metrics cohomologous to $\omega$. If a cscK metric exists in $\mathcal H$, 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 existence of a cscK metric in $\mathcal H$ 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.