LEADER 04856nam 22007455 450 001 996466663503316 005 20200701010914.0 010 $a3-642-23669-3 024 7 $a10.1007/978-3-642-23669-3 035 $a(CKB)3390000000021680 035 $a(SSID)ssj0000609363 035 $a(PQKBManifestationID)11433923 035 $a(PQKBTitleCode)TC0000609363 035 $a(PQKBWorkID)10625703 035 $a(PQKB)11413119 035 $a(DE-He213)978-3-642-23669-3 035 $a(MiAaPQ)EBC3070395 035 $a(PPN)159084725 035 $a(EXLCZ)993390000000021680 100 $a20120104d2012 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aComplex Monge?Ampère Equations and Geodesics in the Space of Kähler Metrics$b[electronic resource] /$fedited by Vincent Guedj 205 $a1st ed. 2012. 210 1$aBerlin, Heidelberg :$cSpringer Berlin Heidelberg :$cImprint: Springer,$d2012. 215 $a1 online resource (VIII, 310 p. 4 illus.) 225 1 $aLecture Notes in Mathematics,$x0075-8434 ;$v2038 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a3-642-23668-5 320 $aIncludes bibliographical references. 327 $a1.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. 330 $aThe purpose of these lecture notes is to provide an introduction to the theory of complex Monge?Ampère operators (definition, regularity issues, geometric properties of solutions, approximation) on compact Kähler manifolds (with or without boundary). These operators are of central use in several fundamental problems of complex differential geometry (Kähler?Einstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after Bedford?Taylor), Monge?Ampère foliations and laminated currents, polynomial hulls and Perron envelopes with no analytic structure, a self-contained presentation of Krylov regularity results, a modernized proof of the Calabi?Yau theorem (after Yau and Kolodziej), an introduction to infinite dimensional riemannian geometry, geometric structures on spaces of Kähler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity theory of Caffarelli?Kohn?Nirenberg?Spruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after Phong?Sturm and Berndtsson). Each chapter can be read independently and is based on a series of lectures by R. Berman, Z. Blocki, S. Boucksom, F. Delarue, R. Dujardin, B. Kolev and A. Zeriahi, delivered to non-experts. The book is thus addressed to any mathematician with some interest in one of the following fields, complex differential geometry, complex analysis, complex dynamics, fully non-linear PDE's and stochastic analysis. 410 0$aLecture Notes in Mathematics,$x0075-8434 ;$v2038 606 $aFunctions of complex variables 606 $aDifferential geometry 606 $aPartial differential equations 606 $aAlgebraic geometry 606 $aSeveral Complex Variables and Analytic Spaces$3https://scigraph.springernature.com/ontologies/product-market-codes/M12198 606 $aDifferential Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M21022 606 $aPartial Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12155 606 $aAlgebraic Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M11019 608 $aAufsatzsammlung. 615 0$aFunctions of complex variables. 615 0$aDifferential geometry. 615 0$aPartial differential equations. 615 0$aAlgebraic geometry. 615 14$aSeveral Complex Variables and Analytic Spaces. 615 24$aDifferential Geometry. 615 24$aPartial Differential Equations. 615 24$aAlgebraic Geometry. 676 $a515/.7242 686 $aSI 850$2rvk 686 $aMAT 146f$2stub 686 $aMAT 322f$2stub 686 $aMAT 354f$2stub 686 $aMAT 537f$2stub 686 $a510$2sdnb 702 $aGuedj$b Vincent$4edt$4http://id.loc.gov/vocabulary/relators/edt 906 $aBOOK 912 $a996466663503316 996 $aComplex Monge?Ampère equations and geodesics in the space of Kähler metrics$91417443 997 $aUNISA