Series 
Lecture notes in mathematics, 00758434 ; 2038 

Lecture notes in mathematics (SpringerVerlag) ;
2038.

Subject 
MongeAmpère equations.


Geodesics (Mathematics)


Kählerian structures.

Alt Name 
Guedj, Vincent.

Description 
1 online resource (viii, 310 pages). 

polychrome rdacc 
Bibliography Note 
Includes bibliographical references. 
Summary 
Annotation 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 byR. 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. 
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. 
ISBN 
9783642236693 (electronic bk.) 

3642236693 (electronic bk.) 

3642236685 (print) 

9783642236686 (print) 

9783642236686 
ISBN/ISSN 
10.1007/9783642236693 
OCLC # 
773291552 
Additional Format 
Printed edition: 9783642236686 
