03731nam a22005175i 4500001001800000003000900018005001700027007001500044008004100059020001800100024003500118050001200153072001700165072002300182082001500205100003100220245011500251264004200366300004400408336002600452337002600478338003600504347002400540490003500564505043800599520165001037650001702687650002702704650002802731650001402759650002002773650003602793650001402829650001702843650005102860650001402911650004202925650001402967700002802981700003103009710003403040773002003074776003603094830003503130856004803165978-0-8176-4622-6DE-He21320180115171437.0cr nn 008mamaa110518s2011 xxu| s |||| 0|eng d a97808176462267 a10.1007/978-0-8176-4622-62doi 4aQA331.7 7aPBKD2bicssc 7aMAT0340002bisacsh04a515.942231 aGreene, Robert E.eauthor.14aThe Geometry of Complex Domainsh[electronic resource] /cby Robert E. Greene, Kang-Tae Kim, Steven G. Krantz. 1aBoston :bBirkhäuser Boston,c2011. aXIV, 303 p. 14 illus.bonline resource. atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda1 aProgress in Mathematics ;v2910 aPreface -- 1 Preliminaries -- 2 Riemann Surfaces and Covering Spaces -- 3 The Bergman Kernel and Metric -- 4 Applications of Bergman Geometry -- 5 Lie Groups Realized as Automorphism Groups -- 6 The Significance of Large Isotropy Groups -- 7 Some Other Invariant Metrics -- 8 Automorphism Groups and Classification of Reinhardt Domains -- 9 The Scaling Method, I -- 10 The Scaling Method, II -- 11 Afterword -- Bibliography -- Index. aThe geometry of complex domains is a subject with roots extending back more than a century, to the uniformization theorem of Poincaré and Koebe and the resulting proof of existence of canonical metrics for hyperbolic Riemann surfaces. In modern times, developments in several complex variables by Bergman, Hörmander, Andreotti-Vesentini, Kohn, Fefferman, and others have opened up new possibilities for the unification of complex function theory and complex geometry. In particular, geometry can be used to study biholomorphic mappings in remarkable ways. This book presents a complete picture of these developments. Beginning with the one-variable case—background information which cannot be found elsewhere in one place—the book presents a complete picture of the symmetries of domains from the point of view of holomorphic mappings. It describes all the relevant techniques, from differential geometry to Lie groups to partial differential equations to harmonic analysis. Specific concepts addressed include: covering spaces and uniformization; Bergman geometry; automorphism groups; invariant metrics; the scaling method. All modern results are accompanied by detailed proofs, and many illustrative examples and figures appear throughout. Written by three leading experts in the field, The Geometry of Complex Domains is the first book to provide systematic treatment of recent developments in the subject of the geometry of complex domains and automorphism groups of domains. A unique and definitive work in this subject area, it will be a valuable resource for graduate students and a useful reference for researchers in the field. 0aMathematics. 0aMathematical analysis. 0aAnalysis (Mathematics). 0aDynamics. 0aErgodic theory. 0aFunctions of complex variables. 0aGeometry.14aMathematics.24aSeveral Complex Variables and Analytic Spaces.24aAnalysis.24aDynamical Systems and Ergodic Theory.24aGeometry.1 aKim, Kang-Tae.eauthor.1 aKrantz, Steven G.eauthor.2 aSpringerLink (Online service)0 tSpringer eBooks08iPrinted edition:z9780817641399 0aProgress in Mathematics ;v29140uhttp://dx.doi.org/10.1007/978-0-8176-4622-6