Geometry of K\"ahler Metrics and Foliations by Holomorphic Discs

TL;DR

This paper develops a new partial regularity theory for homogeneous complex Monge-Ampère equations by studying foliations by holomorphic curves, with applications to the uniqueness and existence conditions of extremal Kähler metrics.

Contribution

It introduces a novel partial regularity framework linking holomorphic foliations to complex Monge-Ampère equations, advancing understanding of Kähler geometry.

Findings

Proved uniqueness of extremal Kähler metrics.

Established necessary conditions for existence of extremal Kähler metrics.

Developed a new partial regularity theory for complex Monge-Ampère equations.

Abstract

The purpose of this paper is to establish a completely new partial regularity theory on certain homogeneous complex Monge-Ampere equations. Our partial regularity theory will be obtained by studying foliations by holomorphic curves and and their relations to homogeneous complex Monge-Ampere equations. As applications, we will prove the uniqueness of extremal K\"ahler metrics and give an necessary condition for existence of extremal K\"ahler metrics. Further applications will be discussed in our forthcoming papers.