On the constant scalar curvature K\"ahler metrics, existence results

TL;DR

This paper advances the understanding of constant scalar curvature K"ahler (cscK) metrics by extending estimates to more general equations, proving a key conjecture relating non-existence to destabilized geodesic rays, and linking properness of $K$-energy to existence.

Contribution

It generalizes a priori estimates for cscK equations, proves Donaldson's conjecture under certain conditions, and establishes the smoothness of weak $K$-energy minimizers.

Findings

Non-existence of cscK metrics corresponds to destabilized geodesic rays.

Properness of $K$-energy implies existence of cscK metrics.

Weak minimizers of $K$-energy are smooth.

Abstract

In this paper, we generalize our apriori estimates on cscK(constant scalar curvature K\"ahler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture that the non-existence of cscK metric is equivalent to the existence of a destabilized geodesic ray where the $K$-energy is non-increasing. Moreover, we prove that the properness of $K$-energy in terms of $L^1$ geodesic distance $d_1$ in the space of K\"ahler potentials implies the existence of cscK metric. Finally, we prove that weak minimizers of the $K$-energy in $(\mathcal{E}^1, d_1)$ are smooth.