Global Analysis of Minimal Surfaces
Many properties of minimal surfaces are of a global nature, and this is already true for the results treated in the first two volumes of the treatise. Part I of the present book can be viewed as an extension of these results. For instance, the first two c
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Series editors M. Berger P. de la Harpe F. Hirzebruch N.J. Hitchin L. Hörmander M. Kashiwara A. Kupiainen G. Lebeau F.-H. Lin B.C. Ngô M. Ratner D. Serre Ya.G. Sinai N.J.A. Sloane A.M. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates S.R.S. Varadhan
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Minimal surfaces with a free boundary. (a) Tongue; (b) Cusp; (c) Loop. Courtesy of E. Pitts For further volumes: http://www.springer.com/series/138
Ulrich Dierkes Stefan Hildebrandt Anthony J. Tromba
Global Analysis of Minimal Surfaces Revised and enlarged 2nd edition With 43 Figures and 5 Colour Plates
Ulrich Dierkes Faculty of Mathematics University of Duisburg-Essen Campus Duisburg Forsthausweg 2 47057 Duisburg Germany [email protected]
Stefan Hildebrandt Mathematical Institute University of Bonn Endenicher Allee 60 53115 Bonn Germany
Anthony J. Tromba Department of Mathematics University of California at Santa Cruz Baskin 261B CA 95064 Santa Cruz USA [email protected]
This volume is the third part of a treatise on Minimal Surfaces in the series Grundlehren der mathematischen Wissenschaften. Part One is Vol. 339 ISBN 978-3-642-11697-1, Part Two is Vol. 340 ISBN 978-3-642-11700-8. A 1st edition of the treatise appeared as Vols. 295 and 296 of the same series.
ISSN 0072-7830 e-ISBN 978-3-642-11706-0 ISBN 978-3-642-11705-3 DOI 10.1007/978-3-642-11706-0 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010930924 Mathematics Subject Classification (2010): 49Q05, 53A05, 53A07, 53B20, 35J20, 35J47, 35J50, 35J75, 49Q20, 30C20, 30F60, 58J05 c Springer-Verlag Berlin Heidelberg 1992, 2010 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: VTEX, Vilnius Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
This book is the third volume of a treatise on minimal surfaces consisting of altogether three volumes which can be read and studied independently of each other. The central theme is boundary value problems for minimal surfaces, such as Plateau’s problem. The present treatise forms a greatly extended version of the monograph Minimal Surfaces I, II by U. Dierkes, S. Hildebrandt, A. K¨ uster, and O. Wohlrab, published in 1992, which is often cited in t
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