Classification Theory of Riemann Surfaces
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 backgrou
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Herausgegeben von J. L. Doob· A. Grothendieck· E. Heinz· F. Hirzebruch E. Hopf . H. Hopf . W. Maak . S. MacLane . W. Magnus M. M. Postnikov . F. K. Schmidt . D. S. Scott . K. Stein
Geschäftsführende Herausgeber B. Eckmann und B. L. vart der Waerden
L. Sario . M. Nakai
Classification Theory of Riemann Surfaces
Springer-Verlag Berlin Heidelberg GmbH 1970
Prof. Leo Sario University of California, Los Angeles
Prof. Mitsuru Nakai Nagoya U niversity
Geschäftsführende Herausgeber:
Prof. Dr. B. Bekmann Eidgenössische Technische Hochschule Zürich
Prof. Dr. B. L. van der Waerden Mathematisches Institut der Universität Zürich
ISBN 978-3-642-48271-7 ISBN 978-3-642-48269-4 (eBook) DOI 10.1007/978-3-642-48269-4
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© by Springer-Verlag Berlin Heide1berg 1970. Library of Congress Catalog Card Number 76-96693 Originally published by Springer-Verlag Berlin· Heidelberg 1970 Softcover reprint ofthe hardcover Ist edition 1970
Preface 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. At the end of his dissertation RIEMANN had already referred to the significance of the existence of the Green's function. This aspect gave rise to a generalization which chronologically ran somewhat parallel to the classical type problem: finding tests for the class 0G of parabolic surface
