Description

The purpose of these lecture notes is to provide an introduction to the theory of complex Monge-Amp re operators (definition, regularity issues, geometric properties of solutions, approximation) on compact K hler manifolds (with or without boundary).
These operators are of central use in several fundamental problems of complex differential geometry (K hler-Einstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after Bedford-Taylor), Monge-Amp re foliations and laminated currents, polynomial hulls and Perron envelopes with no analytic structure, a self-contained presentation of Krylov regularity results, a modernized proof of the Calabi-Yau theorem (after Yau and Kolodziej), an introduction to infinite dimensional riemannian geometry, geometric structures on spaces of K hler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity theory of Caffarelli-Kohn-Nirenberg-Spruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after Phong-Sturm and Berndtsson).

Each chapter can be read independently and is based on a series of lectures by R. Berman, Z. Blocki, S. Boucksom, F. Delarue, R. Dujardin, B. Kolev and A. Zeriahi, delivered to non-experts. The book is thus addressed to any mathematician with some interest in one of the following fields, complex differential geometry, complex analysis, complex dynamics, fully non-linear PDE's and stochastic analysis.