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The main theme of this book is the homotopy principle for holomorphic mappings from Stein manifolds to the newly introduced class of Oka manifolds. The book contains the first complete account of Oka-Grauert theory and its modern extensions, initiated by Mikhail Gromov and developed in the last decade by the author and his collaborators. Included is the first systematic presentation of the theory of holomorphic automorphisms of complex Euclidean spaces, a survey on Stein neighborhoods, connections between the geometry of Stein surfaces and Seiberg-Witten theory, and a wide variety of applications ranging from classical to contemporary.
Homotopy theory --- Stein manifolds --- Holomorphic mappings --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Calculus --- Stein manifolds. --- Holomorphic mappings. --- Homotopy theory. --- Deformations, Continuous --- Mappings, Holomorphic --- Mathematics. --- Functions of complex variables. --- Functions of real variables. --- Real Functions. --- Functions of a Complex Variable. --- Topology --- Functions of several complex variables --- Mappings (Mathematics) --- Manifolds (Mathematics) --- Complex variables --- Elliptic functions --- Functions of real variables --- Math --- Science --- Real variables --- Functions of complex variables
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This book, now in a carefully revised second edition, provides an up-to-date account of Oka theory, including the classical Oka-Grauert theory and the wide array of applications to the geometry of Stein manifolds. Oka theory is the field of complex analysis dealing with global problems on Stein manifolds which admit analytic solutions in the absence of topological obstructions. The exposition in the present volume focuses on the notion of an Oka manifold introduced by the author in 2009. It explores connections with elliptic complex geometry initiated by Gromov in 1989, with the Andersén-Lempert theory of holomorphic automorphisms of complex Euclidean spaces and of Stein manifolds with the density property, and with topological methods such as homotopy theory and the Seiberg-Witten theory. Researchers and graduate students interested in the homotopy principle in complex analysis will find this book particularly useful. It is currently the only work that offers a comprehensive introduction to both the Oka theory and the theory of holomorphic automorphisms of complex Euclidean spaces and of other complex manifolds with large automorphism groups.
Mathematics. --- Functions of complex variables. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Functions of a Complex Variable. --- Global Analysis and Analysis on Manifolds. --- Several Complex Variables and Analytic Spaces. --- Stein manifolds. --- Holomorphic mappings. --- Mappings, Holomorphic --- Functions of several complex variables --- Mappings (Mathematics) --- Manifolds (Mathematics) --- Global analysis. --- Differential equations, partial. --- Partial differential equations --- Complex variables --- Elliptic functions --- Functions of real variables --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic
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Functional analysis --- Mathematical analysis --- functies (wiskunde)
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Algebraic geometry --- Differential geometry. Global analysis --- Analytical spaces --- Differential equations --- Mathematical analysis --- Mathematics --- differentiaalvergelijkingen --- analyse (wiskunde) --- topologie (wiskunde) --- complexe veranderlijken --- statistiek --- wiskunde
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None. The Ergebnisse volumes do not have back cover texts.
Functional analysis --- Mathematical analysis --- functies (wiskunde)
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This monograph offers the first systematic treatment of the theory of minimal surfaces in Euclidean spaces by complex analytic methods, many of which have been developed in recent decades as part of the theory of Oka manifolds (the h-principle in complex analysis). It places particular emphasis on the study of the global theory of minimal surfaces with a given complex structure. Advanced methods of holomorphic approximation, interpolation, and homotopy classification of manifold-valued maps, along with elements of convex integration theory, are implemented for the first time in the theory of minimal surfaces. The text also presents newly developed methods for constructing minimal surfaces in minimally convex domains of Rn, based on the Riemann-Hilbert boundary value problem adapted to minimal surfaces and holomorphic null curves. These methods also provide major advances in the classical Calabi-Yau problem, yielding in particular minimal surfaces with the conformal structure of any given bordered Riemann surface. Offering new directions in the field and several challenging open problems, the primary audience of the book are researchers (including postdocs and PhD students) in differential geometry and complex analysis. Although not primarily intended as a textbook, two introductory chapters surveying background material and the classical theory of minimal surfaces also make it suitable for preparing Masters or PhD level courses.
Algebraic geometry --- Differential geometry. Global analysis --- Analytical spaces --- Mathematical analysis --- Mathematics --- analyse (wiskunde) --- topologie (wiskunde) --- complexe veranderlijken --- statistiek --- wiskunde --- geometrie
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Riemann, Surfaces de --- Riemann surfaces --- Minimal surfaces --- Surfaces minimales --- Minimal surfaces. --- Sprays (Mathematics) --- Analytic spaces. --- Affine differential geometry. --- Approximation theory. --- Holomorphic mappings. --- Differential geometry {For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx} -- Classical differential geometry -- Minimal surfaces, surfaces with pr --- Several complex variables and analytic spaces {For infinite-dimensional holomorphy, see 46G20, 58B12} -- Local analytic geometry [See also 13-XX and 14-XX] -- Analytic subsets of affine space. --- Several complex variables and analytic spaces {For infinite-dimensional holomorphy, see 46G20, 58B12} -- Holomorphic convexity -- Holomorphic and polynomial approximation, Runge pairs, interpolation. --- Several complex variables and analytic spaces {For infinite-dimensional holomorphy, see 46G20, 58B12} -- Holomorphic mappings and correspondences -- Holomorphic mappings, (holomorphic) embeddings and
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This monograph offers the first systematic treatment of the theory of minimal surfaces in Euclidean spaces by complex analytic methods, many of which have been developed in recent decades as part of the theory of Oka manifolds (the h-principle in complex analysis). It places particular emphasis on the study of the global theory of minimal surfaces with a given complex structure. Advanced methods of holomorphic approximation, interpolation, and homotopy classification of manifold-valued maps, along with elements of convex integration theory, are implemented for the first time in the theory of minimal surfaces. The text also presents newly developed methods for constructing minimal surfaces in minimally convex domains of Rn, based on the Riemann–Hilbert boundary value problem adapted to minimal surfaces and holomorphic null curves. These methods also provide major advances in the classical Calabi–Yau problem, yielding in particular minimal surfaces with the conformal structure of any given bordered Riemann surface. Offering new directions in the field and several challenging open problems, the primary audience of the book are researchers (including postdocs and PhD students) in differential geometry and complex analysis. Although not primarily intended as a textbook, two introductory chapters surveying background material and the classical theory of minimal surfaces also make it suitable for preparing Masters or PhD level courses.
Global analysis (Mathematics). --- Manifolds (Mathematics). --- Functions of complex variables. --- Global Analysis and Analysis on Manifolds. --- Several Complex Variables and Analytic Spaces. --- Complex variables --- Elliptic functions --- Functions of real variables --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Global analysis (Mathematics) --- Manifolds (Mathematics) --- Minimal surfaces. --- Surfaces, Minimal --- Maxima and minima --- Anàlisi global (Matemàtica) --- Funcions de variables complexes --- Varietats (Matemàtica) --- Superfícies mínimes --- Màxims i mínims --- Aplicacions de Gauss --- Anàlisi complexa --- Funció d'una variable complexa --- Variables complexes --- Aplicacions quasiconformes --- Funcions abelianes --- Funcions analítiques --- Funcions convexes --- Funcions de diverses variables complexes --- Funcions enteres --- Funcions meromorfes --- Funcions univalents --- Grups discontinus --- Invariants conformes --- Problemes de contorn --- Sèries de Lie --- Teoria geomètrica de funcions --- Funcions de variables reals --- Funcions el·líptiques --- Sistemes dinàmics complexos --- Topologia diferencial --- Geodèsiques (Matemàtica) --- Geometria espectral --- Teoria del punt crític (Anàlisi matemàtica) --- Varietats de dimensió infinita --- Varietats analítiques --- Geometria diferencial --- Topologia --- Varietats topològiques --- Catàstrofes (Matemàtica) --- Homeomorfismes --- Subvarietats (Matemàtica) --- Topologia de baixa dimensió --- Tor (Geometria) --- Varietats complexes --- Varietats de Calabi-Yau --- Varietats de Grassmann --- Varietats diferenciables --- Varietats de Kähler --- Varietats simplèctiques
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Algebraic geometry --- Differential geometry. Global analysis --- Analytical spaces --- Mathematical analysis --- Mathematics --- analyse (wiskunde) --- topologie (wiskunde) --- complexe veranderlijken --- statistiek --- wiskunde --- geometrie
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