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Grundlehren der mathematischen Wissenschaften

Classification Theory of Riemann Surfaces

Authors: Sario, Leo, Nakai, Mitsuru

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About this book

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.

Table of contents (7 chapters)

Table of contents (7 chapters)

Buy this book

eBook $84.99
price for USA in USD (gross)
  • ISBN 978-3-642-48269-4
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $109.99
price for USA in USD
  • ISBN 978-3-642-48271-7
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Classification Theory of Riemann Surfaces
Authors
Series Title
Grundlehren der mathematischen Wissenschaften
Series Volume
164
Copyright
1970
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin
eBook ISBN
978-3-642-48269-4
DOI
10.1007/978-3-642-48269-4
Softcover ISBN
978-3-642-48271-7
Series ISSN
0072-7830
Edition Number
1
Number of Pages
XX, 450
Topics