A Theory of Branched Minimal Surfaces
One of the most elementary questions in mathematics is whether an area minimizing surface spanning a contour in three space is immersed or not; i.e. does its derivative have maximal rank everywhere. The purpose of this monograph is to present an elementar
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Anthony Tromba
A Theory of Branched Minimal Surfaces
Anthony Tromba Department of Mathematics University of California at Santa Cruz Santa Cruz, CA, USA
ISSN 1439-7382 Springer Monographs in Mathematics ISBN 978-3-642-25619-6 e-ISBN 978-3-642-25620-2 DOI 10.1007/978-3-642-25620-2 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011945164 Mathematics Subject Classification: 30B10, 49J50, 49Q05, 53A10, 58E12, 58C20 © Springer-Verlag Berlin Heidelberg 2012 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. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
It is amazing how far you can come in life if you really know how to differentiate. Jerrold E. Marsden There are no problems in geometry requiring the calculation of a derivative greater than four. S.S. Chern A theorem isn’t really proved until you can put it in a book. Marty Golubitsky
To My Dear Friends Michael Buchner Stefan Hildebrandt Jerry Marsden Fritz Tomi
Preface
This book can be considered a continuation of The Regularity of Minimal Surfaces by Ulrich Dierkes, Stefan Hildebrandt and Anthony Tromba, Volume 340 of the Grundlehren der Mathematischen Wissenchaften. The central theme is the study of branch points for minimal surfaces with the goal of providing a new approach to the elementary question of whether minima of area or energy must be immersed. One of the main difficulties with the current theory of branch points is the transparency and sophistication of the proofs of the main theorems. For example, Osserman’s original 1970 cut and paste proof, that absolute minima are free of interior branch points remains, for the most part, open only to experts. Furthermore, before the appearance of this volume, no complete proof has appeared in one place. In the 1960’s the development of global nonlinear analysis and the idea of doing calculus or analysis on infinite dimensional manifolds had created a great deal of excitement, especially through the pioneering work of Jim Eells, Dick Palais and Steve Smale. The goal of this book is to develop entirely new and elementary methods, in the spirit of global analysis, to address this beautiful
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