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Table of contents

  1. Front Matter
    Pages I-XX
  2. L. Sario, M. Nakai
    Pages 1-9
  3. L. Sario, M. Nakai
    Pages 10-78
  4. L. Sario, M. Nakai
    Pages 79-144
  5. L. Sario, M. Nakai
    Pages 145-221
  6. L. Sario, M. Nakai
    Pages 222-285
  7. L. Sario, M. Nakai
    Pages 286-363
  8. L. Sario, M. Nakai
    Pages 364-390
  9. Back Matter
    Pages 391-449

About this book

Introduction

The purpose of the present monograph is to systematically develop a classification theory of Riemann surfaces. Some first steps will also be taken toward a classification of Riemannian spaces. Four phases can be distinguished in the chronological background: the type problem; general classification; compactifications; and extension to higher dimensions. The type problem evolved in the following somewhat overlapping steps: the Riemann mapping theorem, the classical type problem, and the existence of Green's functions. The Riemann mapping theorem laid the foundation to classification theory: there are only two conformal equivalence classes of (noncompact) simply connected regions. Over half a century of efforts by leading mathematicians went into giving a rigorous proof of the theorem: RIEMANN, WEIERSTRASS, SCHWARZ, NEUMANN, POINCARE, HILBERT, WEYL, COURANT, OSGOOD, KOEBE, CARATHEODORY, MONTEL. The classical type problem was to determine whether a given simply connected covering surface of the plane is conformally equivalent to the plane or the disko The problem was in the center of interest in the thirties and early forties, with AHLFORS, KAKUTANI, KOBAYASHI, P. MYRBERG, NEVANLINNA, SPEISER, TEICHMÜLLER and others obtaining incisive specific results. The main problem of finding necessary and sufficient conditions remains, however, unsolved.

Keywords

Riemannsche Fläche Surfaces function proof theorem

Authors and affiliations

  • L. Sario
    • 1
  • M. Nakai
    • 2
  1. 1.Unversity of CaliforniaCaliforniaUSA
  2. 2.Nagoya UnversityGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-48269-4
  • Copyright Information Springer-Verlag Berlin Heidelberg 1970
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-48271-7
  • Online ISBN 978-3-642-48269-4
  • Series Print ISSN 0072-7830
  • Buy this book on publisher's site
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