Convexity of the K-energy on the space of Kahler metrics and uniqueness of extremal metrics

2014/05/02 by Robert J. Berman, Bo Berndtsson, Berman, Robert J. +1 · 9 citations
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.1405.0401

Abstract

We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kahler potentials on a compact Kahler manifold thus confirming a conjecture of Chen and give some applications in Kahler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogenuous Monge-Ampere equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.

Cited by

• Twisted Calabi functional and twisted Calabi flow
• Convexity of the K-energy and Uniqueness of Extremal metrics -- An Expository Introduction
• Monge-Ampère equations with prescribed singularities on compact Hermitian manifolds
• A YTD correspondence for constant scalar curvature metrics
• On the effective YTD conjecture
• The CR Yamabe invariant and constant scalar curvature Sasaki metrics
• Monge-Ampère operators, energy functionals, and uniqueness of Sasaki-extremal metrics
• Uniform K-stability, Duistermaat-Heckman measures and singularities of\n pairs
• The Calabi flow with rough initial data

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