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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
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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
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
Email: bob@chalmers.se

DOI: https://doi.org/10.1090/jams/880
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

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