04064nam 2200673Ia 450 991013278580332120200520144314.09783642236693364223669310.1007/978-3-642-23669-3(CKB)3390000000021680(SSID)ssj0000609363(PQKBManifestationID)11433923(PQKBTitleCode)TC0000609363(PQKBWorkID)10625703(PQKB)11413119(DE-He213)978-3-642-23669-3(MiAaPQ)EBC3070395(PPN)159084725(EXLCZ)99339000000002168020110730d2012 uy 0engurnn|008mamaatxtccrComplex Monge-Ampere equations and geodesics in the space of Kahler metrics /Vincent Guedj, editor1st ed. 2012.Berlin ;Heidelberg Springer Verlag20121 online resource (VIII, 310 p. 4 illus.) Lecture notes in mathematics ;2038Bibliographic Level Mode of Issuance: Monograph9783642236686 3642236685 Includes bibliographical references.1.Introduction -- I. The Local Homogenious Dirichlet Problem.-2. Dirichlet Problem in Domains of Cn -- 3. Geometric Maximality -- II. Stochastic Analysis for the Monge-Ampère Equation -- 4. Probabilistic Approach to Regularity -- III. Monge-Ampère Equations on Compact Manifolds -- 5.The Calabi-Yau Theorem -- IV Geodesics in the Space of Kähler Metrics -- 6. The Riemannian Space of Kähler Metrics -- 7. MA Equations on Manifolds with Boundary -- 8. Bergman Geodesics.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.Lecture notes in mathematics (Springer-Verlag) ;2038.Monge-Ampère equationsGeodesics (Mathematics)Kählerian structuresMonge-Ampère equations.Geodesics (Mathematics)Kählerian structures.516.362SI 850rvkMAT 146fstubMAT 322fstubMAT 354fstubMAT 537fstub510sdnbGuedj Vincent524796MiAaPQMiAaPQMiAaPQBOOK9910132785803321Complex Monge-Ampere equations and geodesics in the space of Kahler metrics4193166UNINA