Uniqueness of constant scalar curvature K\"ahler metrics with cone singularities, I: Reductivity

TL;DR

This paper proves the uniqueness of conic constant scalar curvature Kähler metrics with cone angles less than π by establishing regularity results and analyzing the conic Lichnerowicz operator.

Contribution

It introduces a new Hölder space for regularity analysis and proves that conic cscK metrics are smoother than initially assumed, also establishing reductivity of the automorphism group.

Findings

Conic cscK metrics are of class C^{4,α,β} when initially C^{2,α,β}.

Reductivity of the automorphism group is established.

New Hölder space C^{4,α,β} is introduced for regularity analysis.

Abstract

The aim of this paper is to investigate uniqueness of conic constant scalar curvature Kaehler (cscK) metrics, when the cone angle is less than $\pi$. We introduce a new H\"older space called $\cC^{4,\a,\b}$ to study the regularities of this fourth order elliptic equation, and prove that any $\cC^{2,\a,\b}$ conic cscK metric is indeed of class $\cC^{4,\a,\b}$. Finally, the reductivity is established by a careful study of the conic Lichnerowicz operator.