Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics (Lecture Notes in Mathematics Book 2038)

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. 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 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-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 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). Pt. I. The Local Homogeneous Dirichlet Problem -- Dirichlet Problem In Domains Of Cn -- Geometric Properties Of Maximal Psh Functions -- Pt. Ii. Stochastic Analysis For The Monge-ampère Equation -- Probabilistic Approach To Regularity -- Pt. Iii. Monge-ampère Equations On Compact Kähler Manifolds -- The Calabi-yau Theorem -- Pt. Iv. Geodesies In The Space Of Kähler Metrics -- The Riemannian Space Of Kähler Metrics -- Monge-ampère Equations On Complex Manifolds With Boundary -- Bergman Geodesics. Vincent Guedj, Editor. Includes Bibliographical References. Annotation The purpose of these lecture notes is to provide an introduction to the theory of complex MongeAmpère 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 (KählerEinstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after BedfordTaylor), MongeAmpère 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 Kähler 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 byR. 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 Front Matter....Pages i-viii Front Matter....Pages 11-11 Introduction....Pages 1-10 Dirichlet Problem in Domains of C n ....Pages 13-32 Geometric Properties of Maximal psh Functions....Pages 33-52 Front Matter....Pages 53-53 Probabilistic Approach to Regularity....Pages 55-198 Front Matter....Pages 199-199 The Calabi–Yau Theorem....Pages 201-227 Front Matter....Pages 229-229 The Riemannian Space of Kähler Metrics....Pages 231-255 Monge–Ampère Equations on Complex Manifolds with Boundary....Pages 257-282 Bergman Geodesics....Pages 283-302 Back Matter....Pages 303-310 Provides an introduction to the theory of complex Monge - Ampere operators on compact Kahler manifolds. This book covers the Dirichlet problem, Monge - Ampere foliations and laminated currents, polynomial hulls and Perron envelopes with no analytic structure, a self-contained presentation of Krylov regularity results, and more.