A variational approach to complex Monge-Ampere equations

Understand

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.