Complex MongeAmpère equations and geodesics in the space of Kähler metrics
 Language
 English.
 Imprint
 Berlin ; New York : Springer, c2012.
 Physical description
 viii, 310 p. : ill. ; 24 cm.
 Series
 Lecture notes in mathematics (SpringerVerlag) 2038.
Access
Available online

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QA3 .L28 V.2038

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QA3 .L28 V.2038
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Contributors
 Contributor
 Guedj, Vincent.
Contents/Summary
 Bibliography
 Includes bibliographical references.
 Contents

 1.Introduction. I. The Local Homogenious Dirichlet Problem.2. Dirichlet Problem in Domains of Cn. 3. Geometric Maximality. II. Stochastic Analysis for the MongeAmpere Equation. 4. Probabilistic Approach to Regularity. III. MongeAmpere Equations on Compact Manifolds. 5.The CalabiYau Theorem. IV Geodesics in the Space of Kahler Metrics. 6. The Riemannian Space of Kahler Metrics. 7. MA Equations on Manifolds with Boundary. 8. Bergman Geodesics.
 (source: Nielsen Book Data)
 Publisher's Summary
 The purpose of these lecture notes is to provide an introduction to the theory of complex MongeAmpere operators (definition, regularity issues, geometric properties of solutions, approximation) on compact Kahler manifolds (with or without boundary). These operators are of central use in several fundamental problems of complex differential geometry (KahlerEinstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after BedfordTaylor), MongeAmpere foliations and laminated currents, polynomial hulls and Perron envelopes with no analytic structure, a selfcontained 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 Kahler 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 nonexperts. 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 nonlinear PDE's and stochastic analysis.
(source: Nielsen Book Data)
Bibliographic information
 Publication date
 2012
 Responsibility
 Vincent Guedj, editor.
 Series
 Lecture notes in mathematics ; 2038
 ISBN
 9783642236686
 3642236685
 9783642236693 (ebook)
 3642236693 (ebook)
 Publisher Number
 Best.Nr.: 80112046