Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics
Journal article, 2017

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.

Plurisubharmonic function

Mabuchi funcional

Constant scalar curvature


Robert Berman

Chalmers, Mathematical Sciences

University of Gothenburg

Bo Berndtsson

University of Gothenburg

Chalmers, Mathematical Sciences, Algebra and geometry

Journal of the American Mathematical Society

0894-0347 (ISSN) 1088-6834 (eISSN)

Vol. 30 4 1165-1196

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