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Riemann Surfaces of Infinite Genus
Joel Feldman, University of British Columbia, Vancouver, BC, Canada, and Horst Knörrer and Eugene Trubowitz, Eidgenössische Technische Hochschule, Zurich, Switzerland
A co-publication of the AMS and Centre de Recherches Mathématiques.
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CRM Monograph Series
2003; 296 pp; hardcover
Volume: 20
ISBN-10: 0-8218-3357-X
ISBN-13: 978-0-8218-3357-5
List Price: US$87
Member Price: US$69.60
Order Code: CRMM/20
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In this book, the authors geometrically construct Riemann surfaces of infinite genus by pasting together plane domains and handles. To achieve a meaningful generalization of the classical theory of Riemann surfaces to the case of infinite genus, one must impose restrictions on the asymptotic behavior of the Riemann surface. In the construction carried out here, these restrictions are formulated in terms of the sizes and locations of the handles and in terms of the gluing maps.

The approach used has two main attractions. The first is that much of the classical theory of Riemann surfaces, including the Torelli theorem, can be generalized to this class. The second is that solutions of Kadomcev-Petviashvilli equations can be expressed in terms of theta functions associated with Riemann surfaces of infinite genus constructed in the book. Both of these are developed here. The authors also present in detail a number of important examples of Riemann surfaces of infinite genus (hyperelliptic surfaces of infinite genus, heat surfaces and Fermi surfaces).

The book is suitable for graduate students and research mathematicians interested in analysis and integrable systems.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

Readership

Graduate students and research mathematicians interested in analysis and integrable systems.

Table of Contents

  • \(L^2\)-cohomology, exhaustions with finite charge and theta series
  • The Torelli Theorem
  • Examples
  • The Kadomcev-Petviashvilli equation
  • Bibliography
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