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

Berman, Robert; Berndtsson, Bo

doi:10.1090/jams/880

Convexity of the $K$-energy on the space of Kähler metrics and uniqueness of extremal metrics
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by Robert J. Berman and Bo Berndtsson

J. Amer. Math. Soc. 30 (2017), 1165-1196

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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