Overview
- The first self contained presentation of Krylov's stochastic analysis for the complex Monge-Ampere equation
- A comprehensive presentation of Yau's proof of the Calabi conjecture
- A great part of the material (both classical results and more recent 4.
- A pedagogical style, lectures accessible to non experts.developments) has not previously appeared in book form
- Written in pedagogicalcal style, lectures accessible to non experts
- Includes supplementary material: sn.pub/extras
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2038)
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Table of contents (8 chapters)
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Front Matter
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The Local Homogeneous Dirichlet Problem
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Front Matter
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The localhomogeneous Dirichlet problem
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Stochastic Analysis for the Monge–Ampère Equation
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Front Matter
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Stochastic analysis for the monge-Ampere equation
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Monge–Ampère Equations on Compact Kähler Manifolds
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Front Matter
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Monge-Ampère equations on compact kahler manifolds
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Geodesics in the Space of Kähler Metrics
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Front Matter
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Geodesics in the space of kahler metrics
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Back Matter
Keywords
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.
Editors and Affiliations
Bibliographic Information
Book Title: Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics
Editors: Vincent Guedj
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-642-23669-3
Publisher: Springer Berlin, Heidelberg
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer-Verlag Berlin Heidelberg 2012
Softcover ISBN: 978-3-642-23668-6Published: 06 January 2012
eBook ISBN: 978-3-642-23669-3Published: 05 January 2012
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: VIII, 310
Number of Illustrations: 4 b/w illustrations
Topics: Several Complex Variables and Analytic Spaces, Differential Geometry, Partial Differential Equations, Algebraic Geometry