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

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

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

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

- DOI https://doi.org/10.1007/978-3-642-23669-3
- Copyright Information Springer-Verlag Berlin Heidelberg 2012
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-642-23668-6
- Online ISBN 978-3-642-23669-3
- Series Print ISSN 0075-8434
- Series Online ISSN 1617-9692
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