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

Robert J. Berman, Bo Berndtsson

Published 2014-05-02, updated 2014-12-02Version 2

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 local Bergman kernels.