On the uniqueness of conical K\"ahler-Einstein metrics

TL;DR

This paper proves the uniqueness of conical Kähler-Einstein metrics assuming the properness of the twisted Ding functional, extending previous work and using approximation techniques.

Contribution

It generalizes prior results by establishing the uniqueness of conical Kähler-Einstein metrics through a new approach involving twisted metrics and approximation.

Findings

Proves uniqueness of conical Kähler-Einstein metrics under properness condition.

Uses approximation of smooth twisted solutions to establish singular metric uniqueness.

Extends previous work to more general conical settings.

Abstract

The purpose of this paper is to prove the uniqueness of conical K\"ahler-Einstein metrics, under the condition that the twisted $Ding$-functional is proper. This is a generalization of the author's previous work, and we shall first investigate the uniqueness of twisted K\"ahler-Einstein metrics, and then use these smooth perturbed solutions to approximate the actual conical one. Finally, it will bring the uniqueness of the limiting singular metric.