The Space of K\"ahler metrics (II)

TL;DR

This paper advances the understanding of the geometry of the space of Kähler metrics, proving it has non-positive curvature and analyzing the gradient flow of the K energy, with implications for the uniqueness of extremal Kähler metrics.

Contribution

It establishes the non-positive curvature of the Kähler metric space and analyzes the gradient flow of the K energy, linking to Donaldson's conjecture.

Findings

The space of Kähler metrics has non-positive curvature.

The gradient flow of the K energy is strictly length decreasing except along automorphisms.

Extremal Kähler metrics are unique up to holomorphic transformations under certain conditions.

Abstract

This paper, the second of a series, deals with the function space of all smooth K\"ahler metrics in any given closed complex manifold $M$ in a fixed cohomology class. The previous result of the second author \cite{chen991} showed that the space is a path length space and it is geodesically convex in the sense that any two points are joined by a unique path, which is always length minimizing and of class C^{1,1}. This already confirms one of Donaldson's conjecture completely and verifies another one partially. In the present paper, we show first of all, that the space is, as expected, a path length space of non-positive curvature in the sense of A. D. Alexanderov. The second result is related to the theory of extremal K\"ahler metrics, namely that the gradient flow of the K energy is strictly length decreasing on all paths except those induced by a path of holomorphic automorphisms of $M$. This result, in particular, implies that extremal K\"ahler metric is unique up to holomorphic transformations, provided that Donaldson's conjecture on the regularity of geodesic is true.