Extremal K\"ahler metrics

TL;DR

This survey reviews recent advances in the study of Calabi's extremal K"ahler metrics, focusing on the Yau-Tian-Donaldson conjecture linking metric existence to stability and exploring scenarios without extremal metrics.

Contribution

It summarizes recent progress on the conjecture and discusses examples where the conjecture holds, providing insights into the existence and non-existence of extremal K"ahler metrics.

Findings

Yau-Tian-Donaldson conjecture relates extremal metrics to stability

Examples where the conjecture has been proven

Discussion on the behavior when extremal metrics do not exist

Abstract

This paper is a survey of some recent progress on the study of Calabi's extremal K\"ahler metrics. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we give some example settings where this conjecture has been established. We then turn to the question of what one expects when no extremal metric exists.