Riemannian Metrics of Constant Mass and Moduli Spaces of Conformal Structures
This monograph deals with recent questions of conformal geometry. It provides in detail an approach to studying moduli spaces of conformal structures, using a new canonical metric for conformal structures. This book is accessible to readers with basic kno
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1743
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Lutz Habermann
Riemannian Metrics of Constant Mass and Moduli Spaces of Conformal Structures
Springer
Author Lutz Habermann Institute of Mathematics and Computer Science Ernst-Moritz-Arndt-University Jahnstr. 15a 17487 Greifswald, Germany E-mail: [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Habermann. Lutz: Riemannian metrics of constant mass and moduli spaces of conformal structures / Lutz Habermann. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer. 2000 (Lecture notes in mathematics; 1743) ISBN 3-540-67987-1
Mathematics Subject Classification (2000): 58D27, 53A30, 53C20 ISSN 0075-8434 ISBN 3-540-67987-1 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version. and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10724216 41/3142-543210 - Printed on acid-free paper
Preface
In understanding spaces of complex structures, it has proved to be useful to construct "canonical" Riemannian metrics on complex manifolds. For such a "canonical" metric 9J, it is required that this metric is uniquely determined by the underlying complex structure J, depends smoothly on that structure, and satisfies f* 9J = 9t: J for any diffeomorphism f of the manifold M under consideration. Moreover, it is expected that gJ has an analyzable behavior as the underlying structure degenerates in some explicit manner. Such a metric 9J for each complex structure J then gives rise to a Riemannian metric on the regular part !JJ1* of the corresponding moduli space !JJ1 by identifying tangent vectors to !JJ1* with harmonic sections of a certain vector bundle on M and taking the L 2-product of such sections with respect to the canonical metric. The best known example of a canonical metric is the hyperbolic metric on compact Riemann surfaces of genus p > 1, whose existence is a consequence of the uniformization theorem. In this case the induced L 2-met ric on the moduli space is the Weil-Petersson metric. The asymptotic behavior of the hyperbolic metric for degenerating Riemann surfaces and its consequences for the geometry of the moduli space was analyzed by Masur [50]. Other examples f
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