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Complex analysis --- Holomorphic functions. --- Complex manifolds. --- Fonctions holomorphes --- Espaces symétriques hermitiens --- 51 <082.1> --- Mathematics--Series --- Espaces symétriques hermitiens --- Complex manifolds --- Holomorphic functions --- Functions, Holomorphic --- Functions of several complex variables --- Analytic spaces --- Manifolds (Mathematics) --- Variétés complexes --- Functions of several complex variables. --- Fonctions de plusieurs variables complexes. --- Stein manifolds. --- Stein, Variétés de --- Plurisubharmonic functions --- Fonctions de plusieurs variables complexes --- Fonctions plurisousharmoniques --- Lie, Groupes de

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The 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.

Analytic functions. --- Functions of complex variables. --- Geometric function theory. --- Geometry, Algebraic. --- Functions of complex variables --- Geometric function theory --- Geometry, Algebraic --- Domains of holomorphy --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Calculus --- Complexes. --- Transformations (Mathematics) --- Linear complexes --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Dynamics. --- Ergodic theory. --- Geometry. --- Several Complex Variables and Analytic Spaces. --- Analysis. --- Dynamical Systems and Ergodic Theory. --- Algorithms --- Differential invariants --- Geometry, Differential --- Algebras, Linear --- Coordinates --- Line geometry --- Differential equations, partial. --- Global analysis (Mathematics). --- Differentiable dynamical systems. --- Euclid's Elements --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Analysis, Global (Mathematics) --- Differential topology --- Partial differential equations --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- 517.1 Mathematical analysis --- Mathematical analysis --- Complex variables --- Elliptic functions --- Functions of real variables

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The 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.

Geometry --- Analytical spaces --- Ergodic theory. Information theory --- Mathematical analysis --- analyse (wiskunde) --- geometrie --- informatietheorie

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This volume includes 28 chapters by authors who are leading researchers of the world describing many of the up-to-date aspects in the field of several complex variables (SCV). These contributions are based upon their presentations at the 10th Korean Conference on Several Complex Variables (KSCV10), held as a satellite conference to the International Congress of Mathematicians (ICM) 2014 in Seoul, Korea. SCV has been the term for multidimensional complex analysis, one of the central research areas in mathematics. Studies over time have revealed a variety of rich, intriguing, new knowledge in complex analysis and geometry of analytic spaces and holomorphic functions which were "hidden" in the case of complex dimension one. These new theories have significant intersections with algebraic geometry, differential geometry, partial differential equations, dynamics, functional analysis and operator theory, and sheaves and cohomology, as well as the traditional analysis of holomorphic functions in all dimensions. This book is suitable for a broad audience of mathematicians at and above the beginning graduate-student level. Many chapters pose open-ended problems for further research, and one in particular is devoted to problems for future investigations.

Mathematics. --- Several Complex Variables and Analytic Spaces. --- Algebraic Geometry. --- Functions of a Complex Variable. --- Differential Geometry. --- Geometry, algebraic. --- Functions of complex variables. --- Differential equations, partial. --- Global differential geometry. --- Mathématiques --- Fonctions d'une variable complexe --- Géométrie différentielle globale --- Integral geometry -- Congresses. --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Integral geometry --- Geometry, Integral --- Algebraic geometry. --- Differential geometry. --- Geometry, Differential --- Complex variables --- Elliptic functions --- Functions of real variables --- Algebraic geometry --- Geometry --- Partial differential equations --- Differential geometry