Convexity of the K-energy and Uniqueness of Extremal metrics -- An Expository Introduction

TL;DR

The paper addresses the problem of uniqueness for constant scalar curvature Kähler metrics in a fixed Kähler class, and extends the framework to extremal metrics. It proves the Mabuchi K-energy $\mathcal{M}$ is convex along weak geodesics in the space of Kähler potentials by a subharmonicity argument grounded in Bergman kernel asymptotics and Monge–Ampère equations, and extends $\mathcal{M}$ to $\mathcal{H}_{1,1}(X,\omega_0)$. The main contributions are establishing full convexity of $\mathcal{M}$ along weak geodesics and deducing uniqueness of cscK metrics up to the automorphism group $\mathrm{Aut}_0(X)$ (and, more generally, uniqueness of extremal metrics up to the reduced automorphism group $G_V$). These results underpin variational approaches to canonical metrics and connect to Yau–Tian–Donaldson-type criteria linking geometric existence to algebro-geometric conditions.

Abstract

This article is an expository introduction to our paper Convexity of the K-energy and Uniqueness of Extremal metrics. We present the main ideas behind the proof that Mabuchi's K-energy functional is convex along weak geodesics in the space of Kahler potentials and explain how this leads to the uniqueness of constant scalar curvature Kahler metrics and extremal metrics up to automorphisms. The emphasis is on the conceptual framework and key techniques.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Corollary 3
• Proposition 4
• Theorem 5
• Theorem 6
• Example 7
• Lemma 8
• Lemma 9
• proof