Date: 22 Sep 2011

Monge–Ampère Equations on Complex Manifolds with Boundary

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Abstract

We survey the proofs of two fundamental results on the resolution of Monge–Ampère equations on complex manifolds with boundary. The first result guarantees the existence of smooth solutions to non-degenerate com- plex Monge–Ampère equations admitting subsolutions, it is a continuation of results due to Caffarelli–Kohn–Nirenberg–Spruck. The second result shows the existence of almost C2 solutions to degenerate complex Monge–Ampère equations admitting subsolutions and yields as a special case X.X.Chen’s result on the existence of almost C2 geodesics in spaces of Kähler metrics.