Convexity of the $K$-energy on the space of Kähler metrics and uniqueness of extremal metrics
Authors:
Robert J. Berman and Bo Berndtsson
Journal:
J. Amer. Math. Soc. 30 (2017), 1165-1196
MSC (2010):
Primary 32Q15, 53C55
DOI:
https://doi.org/10.1090/jams/880
Published electronically:
March 2, 2017
MathSciNet review:
3671939
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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.
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Additional Information
Robert J. Berman
Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Gothenburg, Sweden
MR Author ID:
743613
Email:
robertb@chalmers.se
Bo Berndtsson
Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Gothenburg, Sweden
MR Author ID:
35620
Email:
bob@chalmers.se
Keywords:
Constant scalar curvature,
Mabuchi funcional,
plurisubharmonic function
Received by editor(s):
December 2, 2014
Received by editor(s) in revised form:
November 16, 2016
Published electronically:
March 2, 2017
Additional Notes:
The first author was supported by grants from the ERC (Euoropean Research Council) and the KAW (Knut and Alice Wallenberg foundation).
The second author was supported by a grant from VR (Vetenskapsrådet)
Article copyright:
© Copyright 2017
American Mathematical Society