Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics

Volume 2038 of the series Lecture Notes in Mathematics pp 257-282


Monge–Ampère Equations on Complex Manifolds with Boundary

  • Sébastien BoucksomAffiliated withCNRS et Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie Email author 

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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.