Geodesics in the space of Kahler cone metrics, II. Uniqueness of constant scalar curvature Kahler cone metrics

TL;DR

This paper advances the understanding of Kahler cone metrics by constructing geodesics, analyzing asymptotic behavior of cscK cone metrics, and proving their uniqueness up to automorphisms, thereby enriching the geometric theory of these metrics.

Contribution

It introduces weighted function spaces, constructs cone geodesics, analyzes asymptotics of cscK cone metrics, and proves their uniqueness up to automorphisms.

Findings

Constructed geodesics in the space of Kahler cone metrics.

Determined detailed asymptotic behavior of cscK cone metrics.

Proved the uniqueness of cscK cone metrics up to automorphisms.

Abstract

This is a continuation of the previous articles on Kahler cone metrics. In this article, we introduce weighted function spaces and provide a self-contained treatment on cone angles in the whole interval $(0,1]$. We first construct geodesics in the space of Kahler cone metrics (cone geodesics). We next determine the very detailed asymptotic behaviour of constant scalar curvature Kahler (cscK) cone metrics, which leads to the reductivity of the automorphism group. Then we establish the linear theory for the Lichnerowicz operator, which immediately implies the openness of the path deforming the cone angles of cscK cone metrics. Finally, we address the problem on the uniqueness of cscK cone metrics and show that the cscK cone metric is unique up to automorphisms.