000 04099nam a22005055i 4500
001 978-3-642-23669-3
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007 cr nn 008mamaa
008 120104s2012 gw | s |||| 0|eng d
020 _a9783642236693
_9978-3-642-23669-3
024 7 _a10.1007/978-3-642-23669-3
_2doi
050 4 _aQA331.7
072 7 _aPBKD
_2bicssc
072 7 _aMAT034000
_2bisacsh
072 7 _aPBKD
_2thema
082 0 4 _a515.94
_223
245 1 0 _aComplex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics
_h[electronic resource] /
_cedited by Vincent Guedj.
264 1 _aBerlin, Heidelberg :
_bSpringer Berlin Heidelberg,
_c2012.
300 _aVIII, 310 p. 4 illus.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v2038
505 0 _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.
520 _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.
650 0 _aDifferential equations, partial.
650 0 _aGlobal differential geometry.
650 0 _aGeometry, algebraic.
650 1 4 _aSeveral Complex Variables and Analytic Spaces.
_0http://scigraph.springernature.com/things/product-market-codes/M12198
650 2 4 _aDifferential Geometry.
_0http://scigraph.springernature.com/things/product-market-codes/M21022
650 2 4 _aPartial Differential Equations.
_0http://scigraph.springernature.com/things/product-market-codes/M12155
650 2 4 _aAlgebraic Geometry.
_0http://scigraph.springernature.com/things/product-market-codes/M11019
700 1 _aGuedj, Vincent.
_eeditor.
_4edt
_4http://id.loc.gov/vocabulary/relators/edt
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9783642236686
776 0 8 _iPrinted edition:
_z9783642236709
830 0 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v2038
856 4 0 _uhttps://doi.org/10.1007/978-3-642-23669-3
912 _aZDB-2-SMA
912 _aZDB-2-LNM
999 _c10831
_d10831