A variational approach to complex Monge-Ampere equations

2009/07/27 by Berman, R. J., Boucksom, S., Guedj, V. +1 · 10 citations
#32W20 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.0907.4490

Abstract

We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kaehler manifold can be solved using a variational method independent of Yau's theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kaehler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kaehler metrics and Berndtsson's positivity of direct images we extend Ding-Tian's variational characterization and Bando-Mabuchi's uniqueness result to singular Kaehler-Einstein metrics. Finally using our variational characterization we prove the existence, uniqueness and convergence of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy in our sense.

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• Monge-Ampère equations with prescribed singularities on compact Hermitian manifolds
• Two Calabi-Yau theorems for degenerations of compact Kähler manifolds
• A YTD correspondence for constant scalar curvature metrics
• A transcendental non-Archimedean Calabi--Yau Theorem with applications to the cscK problem
• A new approach to the Monge-Ampère eigenvalue problem
• Lines in the space of Kähler metrics
• An Iterative Approach to the Complex Monge-Ampère Eigenvalue Problem

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