An Introduction to Riemann Surfaces
This textbook presents a unified approach to compact and noncompact Riemann surfaces from the point of view of the L² -method, a powerful technique used in the theory of several complex variables. The work features a simple construction of a strictl
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Terrence Napier Mohan Ramachandran
An Introduction to Riemann Surfaces
Terrence Napier Department of Mathematics Lehigh University Bethlehem, PA 18015 USA [email protected]
Mohan Ramachandran Department of Mathematics SUNY at Buffalo Buffalo, NY 14260 USA [email protected]
ISBN 978-0-8176-4692-9 e-ISBN 978-0-8176-4693-6 DOI 10.1007/978-0-8176-4693-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011936871 Mathematics Subject Classification (2010): 14H55, 30Fxx, 32-01 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com)
To Raghavan Narasimhan
Preface
A Riemann surface X is a connected 1-dimensional complex manifold, that is, a connected Hausdorff space that is locally homeomorphic to open subsets of C with complex analytic coordinate transformations (this also makes X a real 2-dimensional smooth manifold). Although a connected second countable real 1-dimensional smooth manifold is simply a line or a circle (up to diffeomorphism), the real 2-dimensional character of Riemann surfaces makes for a much more interesting topological characterization and a still more interesting complex analytic characterization. A noncompact (i.e., an open) Riemann surface satisfies analogues of the classical theorems of complex analysis, for example the Mittag-Leffler theorem, the Weierstrass theorem, and the Runge approximation theorem (this development began only in the 1940s with the work of, for example, Behnke and Stein). On the other hand, by the maximum principle, a compact Riemann surface admits only constant holomorphic functions. However, compact Riemann surfaces do admit a great many meromorphic functions. This property leads to powerful theorems, in particular, the crucial Riemann–Roch theorem. The theory of Riemann surfaces occupies a unique position in modern mathematics, lying at the intersection of analysis, algebra, geometry, and topology. Most earlier books on this subject have tended to focus on its algebraic-geometric and number-theoretic aspects, rather than its analytic aspects. This book takes the point of view that Riemann surface theory lies at the root of much of modern analysis, and it exploits t
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