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

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.

v1: 26 pages, v2: 31 pages (improved exposition and a new title)v3 minor revision

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