Uniqueness of constant scalar curvature Kähler metrics with cone singularities, I: Reductivity

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 Hölder 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.

37 pages, typos corrected, a new subsection added to explain a global definition of C^{4,\a,\b} space

Cited by in corpus (7)

• Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor
• Construction of constant scalar curvature Kähler cone metrics
• Expansion formula for complex Monge-Ampère equation along cone singularities
• Existence of constant scalar curvature Kaehler cone metrics, properness and geodesic stability
• Geodesics in the space of Kahler cone metrics, II. Uniqueness of constant scalar curvature Kahler cone metrics
• Generalized Matsushima's theorem and Kähler-Einstein cone metrics
• Invariant scalar-flat Kähler metrics on line bundles over generalized flag varieties