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

Journal article
Authors Robert Berman
Bo Berndtsson
Published in Journal of the American Mathematical Society
Volume 30
Issue 4
Pages 1165-1196
ISSN 0894-0347
Publication year 2017
Published at Department of Mathematical Sciences
Pages 1165-1196
Language en
Links doi.org/10.1090/jams/880
Keywords scalar curvature, monge-ampere, projective embeddings, einstein metrics, ricci solitons, manifolds, geometry, stability, existence, geodesics
Subject categories Mathematics

Abstract

We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kähler potentials on a compact Kähler manifold, thus confirming a conjecture of Chen, and give some applications in Kähler 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 homogeneous Monge-Ampère equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.

Page Manager: Webmaster|Last update: 9/11/2012
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