Complex MongeAmpre Equations and Geodesics in the Space of Khler Metrics

The purpose of these lecture notes is to provide an introduction to the theory of complex MongeAmpre operators (definition, regularity issues, geometric properties of solutions, approximation) on compact Khler manifolds (with or without boundary). These operators are of central use in several fundamental problems of complex differential geometry (KhlerEinstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after BedfordTaylor), MongeAmpre 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 CalabiYau theorem (after Yau and Kolodziej), an introduction to infinite dimensional riemannian geometry, geometric structures on spaces of Khler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity theory of CaffarelliKohnNirenbergSpruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after PhongSturm 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.