Minimal Surfaces from a Complex Analytic Viewpoint (Paperback)

Antonio Alarcon

ISBN 10: 303069058X / ISBN 13: 9783030690588

Editore: Springer Nature Switzerland AG, Cham, 2022

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Paperback. 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 RiemannHilbert boundary value problem adapted to minimal surfaces and holomorphic null curves. These methods also provide major advances in the classical CalabiYau 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. 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). Shipping may be from multiple locations in the US or from the UK, depending on stock availability.

Codice articolo 9783030690588

Dati bibliografici

• Titolo: Minimal Surfaces from a Complex Analytic Viewpoint (Paperback)
• Editore: Springer Nature Switzerland AG, Cham
• Data di pubblicazione: 2022
• Lingua: Inglese
• ISBN 10: 303069058X
• ISBN 13: 9783030690588
• Rilegatura: Paperback
• Condizione: new

Descrizione articolo

Riassunto

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.

Informazioni sull'autore

The research interests of Antonio Alarcón lie primarily in minimal surfaces, Riemann surfaces, complex geometry, and holomorphic contact geometry. His main results contribute to the study of the global theory of minimal surfaces in Euclidean spaces by using both classical and modern complex analytic methods. He also studied Bryant surfaces in the hyperbolic space, complex curves and hypersurfaces in complex Euclidean spaces, and holomorphic Legendrian curves in complex contact manifolds.

Franc Forstneric is a complex analyst who has made major contributions to the Cauchy-Riemann geometry, the theory and applications of holomorphic automorphisms of complex Euclidean spaces and related complex manifolds with large automorphism groups, and modern Oka theory, focusing on solving nonlinear complex analytic problems in the absence of topological obstructions. In the last decade he has been applying complex analytic methods to minimal surfaces and holomorphic contact geometry.

Francisco J. López is a specialist in the global theory of minimal surfaces in complete flat manifolds, Riemann surfaces, and complex analysis. Among his main contributions are those concerning the Gauss map of minimal surfaces, the geometry of complete minimal surfaces of finite total curvature, the Calabi–Yau problem, approximation and interpolation theory, and global geometry of complex curves.