A variational approach to complex Monge-Ampere equations

TL;DR

This paper introduces a variational method to solve degenerate complex Monge-Ampere equations on compact Kaehler manifolds, extending classical results and applying to Kähler-Einstein metrics and balanced metrics.

Contribution

It develops a new variational approach independent of Yau's theorem for solving complex Monge-Ampere equations and extends key results to singular Kähler-Einstein metrics and balanced metrics.

Findings

Solved degenerate Monge-Ampere equations via variational methods.

Extended Ding-Tian and Bando-Mabuchi results to singular Kähler-Einstein metrics.

Proved existence, uniqueness, and convergence of balanced metrics.

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.