Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics

  • Vincent Guedj

Part of the Lecture Notes in Mathematics book series (LNM, volume 2038)

Table of contents

  1. Front Matter
    Pages i-viii
  2. The Local Homogeneous Dirichlet Problem

    1. Front Matter
      Pages 11-11
    2. Vincent Guedj
      Pages 1-10
    3. Vincent Guedj, Ahmed Zeriahi
      Pages 13-32
    4. Romain Dujardin, Vincent Guedj
      Pages 33-52
  3. Stochastic Analysis for the Monge–Ampère Equation

    1. Front Matter
      Pages 53-53
    2. François Delarue
      Pages 55-198
  4. Monge–Ampère Equations on Compact Kähler Manifolds

    1. Front Matter
      Pages 199-199
    2. Zbigniew Błocki
      Pages 201-227
  5. Geodesics in the Space of Kähler Metrics

    1. Front Matter
      Pages 229-229
    2. Boris Kolev
      Pages 231-255
    3. Robert Berman, Julien Keller
      Pages 283-302
  6. Back Matter
    Pages 303-310

About this book


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.


32-XX, 53-XX, 35-XX, 14-XX Complex Monge-Ampere equations Geodesics in the space of Kaehler metrics Kaehler metrics Stochastic analysis

Editors and affiliations

  • Vincent Guedj
    • 1
  1. 1.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouseFrance

Bibliographic information