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The purpose of these lecture notes is to provide an introduction to the theory of complex Monge–Ampère operators (definition, regularity issues, geometric properties of solutions, approximation) on compact Kähler manifolds (with or without boundary). These operators are of central use in several fundamental problems of complex differential geometry (Kähler–Einstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after Bedford–Taylor), Monge–Ampère 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 Kähler 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.

Monge-Ampáere equations --- Geodesics (Mathematics) --- Kèahlerian structures --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Mathematical Theory --- Calculus --- Monge-Ampère equations. --- Kählerian structures. --- Structures, Kählerian --- Equations, Monge-Ampère --- Mathematics. --- Algebraic geometry. --- Partial differential equations. --- Functions of complex variables. --- Differential geometry. --- Several Complex Variables and Analytic Spaces. --- Differential Geometry. --- Partial Differential Equations. --- Algebraic Geometry. --- Differential geometry --- Complex variables --- Elliptic functions --- Functions of real variables --- Partial differential equations --- Algebraic geometry --- Math --- Science --- Complex manifolds --- Geometry, Differential --- Hermitian structures --- Global analysis (Mathematics) --- Differential equations, Partial --- Differential equations, partial. --- Global differential geometry. --- Geometry, algebraic. --- Monge-Ampere equations. --- Kahlerian structures.

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Differential geometry. Global analysis --- Geometry --- Analytical spaces --- Partial differential equations --- Mathematical analysis --- differentiaalvergelijkingen --- landmeetkunde --- differentiaal geometrie

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This volume collects lecture notes from courses offered at several conferences and workshops, and provides the first exposition in book form of the basic theory of the Kähler-Ricci flow and its current state-of-the-art. While several excellent books on Kähler-Einstein geometry are available, there have been no such works on the Kähler-Ricci flow. The book will serve as a valuable resource for graduate students and researchers in complex differential geometry, complex algebraic geometry and Riemannian geometry, and will hopefully foster further developments in this fascinating area of research.   The Ricci flow was first introduced by R. Hamilton in the early 1980s, and is central in G. Perelman’s celebrated proof of the Poincaré conjecture. When specialized for Kähler manifolds, it becomes the Kähler-Ricci flow, and reduces to a scalar PDE (parabolic complex Monge-Ampère equation). As a spin-off of his breakthrough, G. Perelman proved the convergence of the Kähler-Ricci flow on Kähler-Einstein manifolds of positive scalar curvature (Fano manifolds). Shortly after, G. Tian and J. Song discovered a complex analogue of Perelman’s ideas: the Kähler-Ricci flow is a metric embodiment of the Minimal Model Program of the underlying manifold, and flips and divisorial contractions assume the role of Perelman’s surgeries.

Mathematics --- Physical Sciences & Mathematics --- Calculus --- Mathematics. --- Partial differential equations. --- Functions of complex variables. --- Differential geometry. --- Several Complex Variables and Analytic Spaces. --- Partial Differential Equations. --- Differential Geometry. --- Kählerian structures. --- Ricci flow. --- Flow, Ricci --- Evolution equations --- Global differential geometry --- Structures, Kählerian --- Complex manifolds --- Geometry, Differential --- Hermitian structures --- Differential equations, partial. --- Global differential geometry. --- Partial differential equations --- Differential geometry --- Complex variables --- Elliptic functions --- Functions of real variables