Overview
- The first book treating minimal surfaces in Euclidean spaces by complex analytic methods, many of which have been developed only recently
- Treats the global theory of minimal surfaces with a given complex structure, providing new results
- Offers new directions in the theory of minimal surfaces and exposes several challenging open problems
Part of the book series: Springer Monographs in Mathematics (SMM)
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Table of contents (9 chapters)
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Front Matter
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Back Matter
Keywords
About this book
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.
Reviews
Authors and Affiliations
About the authors
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 Forstnerič 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.
Bibliographic Information
Book Title: Minimal Surfaces from a Complex Analytic Viewpoint
Authors: Antonio Alarcón, Franc Forstnerič, Francisco J. López
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-3-030-69056-4
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
Hardcover ISBN: 978-3-030-69055-7Published: 11 March 2021
Softcover ISBN: 978-3-030-69058-8Published: 12 March 2022
eBook ISBN: 978-3-030-69056-4Published: 10 March 2021
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XIII, 430
Number of Illustrations: 3 b/w illustrations, 21 illustrations in colour
Topics: Global Analysis and Analysis on Manifolds, Several Complex Variables and Analytic Spaces