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

TL;DR

This paper explains how the convexity of the K-energy functional in Kahler geometry ensures the uniqueness of constant scalar curvature and extremal metrics, emphasizing the core ideas and techniques involved.

Contribution

It provides an accessible exposition of the proof that the K-energy is convex along weak geodesics, leading to metric uniqueness results.

Findings

K-energy functional is convex along weak geodesics

Convexity implies uniqueness of extremal metrics

Highlights key techniques in the proof

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.