The purpose of these lecture notes is to provide an introduction to the theory of complex
Monge-Ampere operators (definition regularity issues geometric properties of solutions
approximation) on compact Kaehler manifolds (with or without boundary). These operators are of
central use in several fundamental problems of complex differential geometry (Kaehler-Einstein
equation uniqueness of constant scalar curvature metrics) complex analysis and dynamics. The
topics covered include. the Dirichlet problem (after Bedford-Taylor). Monge-Ampere 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 Kaehler metrics (after Mabuchi Semmes and Donaldson).
generalizations of the regularity theory of Caffarelli-Kohn-Nirenberg-Spruck (after Guan Chen
and Blocki). Bergman approximation of geodesics (after Phong-Sturm and Berndtsson) Each chapter
can be read independently and is based on a series of lectures 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.
stochastic analysis. lar curvature metrics) complex analysis and dynamics. The topics covered
include. the Dirichlet problem (after Bedford-Taylor). Monge-Ampere 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 Kaehler metrics (after Mabuchi Semmes and Donaldson). generalizations
of the regularity theory of Caffarelli-Kohn-Nirenberg-Spruck (after Guan Che
