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

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.

To appear in the Proceedings of the International Conference of Basic Science 2025