http://dx.doi.org/10.1090/stml/013 Selected Titles in This Series 13 Frederick J. Almgren, Jr., Plateau's problem: An invitation to varifold geometry, Revised edition, 2001 12 W. J. Kaczor and M. T. Nowak, Problems in mathematical analysis II: Continuity and differentiation, 2001 11 Michael Mesterton-Gibbons, An introduction to game-theoretic modelling, 2000 10 John Oprea, The mathematics of soap films: Explorations with Maple®, 2000 9 David E. Blair, Inversion theory and conformal mapping, 2000 8 Edward B. Burger, Exploring the number jungle: A journey into diophantine analysis, 2000 7 Judy L. Walker, Codes and curves, 2000 6 Gerald Tenenbaum and Michel Mendes France, The prime numbers and their distribution, 2000 5 Alexander Mehlmann, The game's afoot! Game theory in myth and paradox, 2000 4 W. J. Kaczor and M. T. Nowak, Problems in mathematical analysis I: Real numbers, sequences and series, 2000 3 Roger Knobel, An introduction to the mathematical theory of waves, 2000 2 Gregory F. Lawler and Lester N. Coyle, Lectures on contemporary probability, 1999 1 Charles Radin, Miles of tiles, 1999 This page intentionally left blank Plateau's Problem An Invitation to Varifold Geometry Revised Edition This page intentionally left blank STUDENT MATHEMATICAL LIBRARY Volume 13 Plateau's Problem An Invitation to Varifold Geometry Revised Edition Frederick J. Almgren, Jr. With new illustrations by Kenneth J. Brakke and John M. Sullivan AMERICAN MATHEMATICAL SOCIETY Editorial Board David Bressoud Carl Pomerance Robert Devaney, Chair Hung-Hsi Wu 2000 Mathematics Subject Classification. Primary 49-01, 26-01, 28-01, 28A75, 49Q15, 49Q20, 58E12. Figures 1-2, 1-3, 1-4, 1-5, 1-6, 1-7, 1-8, l-9a, l-9b, 1-11, l-12a, l-12b, 1-13, l-14a, l-14b, 1-15, 2-6, 2-7, 4-6, 4-7, and 4-8 appear in this volume courtesy of Kenneth A. Brakke and John M. Sullivan. Library of Congress Cataloging-in-Publication Data Almgren, Frederick J. Plateau's problem : an invitation to varifold geometry / Frederick J. Alm• gren, Jr. p. cm. — (Student mathematical library, ISSN 1520-9121 ; v. 13) Originally published: New York : W. A. Benjamin, 1966, in series: Mathe• matics monograph series. Includes bibliographical references and index. ISBN 0-8218-2747-2 (alk. paper) 1. Minimal surfaces. 2. Plateau's problem. 3. Differential topology. I. Ti• tle. II. Series. QA644.A4 2001 516.3/62—dc21 2001022082 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected]. © 1966 held by Jean Taylor. All rights reserved. Reprinted with corrections by the American Mathematical Society, 2001. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 06 05 04 03 02 01 Contents Foreword to the AMS Edition ix Editors' Foreword xiii Preface xv Chapter 1. The Phenomena of Least Area Problems 1 Chapter 2. Integration of Differential Forms over Rectifiable Sets 15 2-1. Notation 15 2-2. Hausdorff measure 16 2-3. The Grassmann algebra and its dual 19 2-4. Differential forms 22 2-5. The Grassmann manifolds associated with it!3 23 2-6. Integration of differential forms over manifolds 25 2-7. Rectifiable sets 31 2-8. Integration of differential forms over rectifiable sets 35 Chapter 3. Varifolds 37 3-1. Rectifiable sets regarded as real valued functions on the space of differential forms 37 Vll vm Contents 3-2. Rectifiable geometry, current geometry, and varifold geometry 39 3-3. Varifolds 44 3-4. The weight of a varifold 48 3-5. Elementary varifolds 52 3-6. Mappings of varifolds 53 Chapter 4. Variational Problems Involving Varifolds 55 4-1. Vector fields and deformations 55 4-2. Variations 56 4-3. Boundaries 58 4-4. Curvature and mean curvature 62 4-5. The compactness theorem for regular integral varifolds 67 4-6. A solution to the existence portion of Plateau's problem 69 4-7. Useful facts about varifolds 71 References 73 Additional References 75 Index 77 Foreword to the AMS Edition I am pleased that Fred's little book is being republished. Many math• ematicians have told me of reading it with pleasure when they were young, and now new generations of mathematicians can share that pleasure. I think Fred would have been pleased, too, and it is too bad that a terrible illness called myelodysplastic syndrome snuck up on him and carried him away at age 63 before he could enjoy this and other tributes. Fred had two quite different sides to his mathematical personal• ity. His research papers tended to be long, complicated affairs that are difficult to read. The wealth of detail could easily overwhelm the reader. In fact, his magnum opus, where he proved that the singular set of area-minimizing integral currents is of codimension 2, runs to over 955 pages of small type [5]. So he made great efforts to present the essence of his theorems, with respect to both the difficulties pre• sented by the problem and the means to surmount them, in the talks he gave and the expository papers he wrote. Since he believed quite strongly that one should never say or write anything that was not precisely mathematically correct, this was no mean feat. This little book "Plateau's Problem" was one of his early gems of exposition. IX X Foreword to the AMS Edition Fred wrote this book shortly after he proved its main theorem and before it had been digested by the mathematical community. It was originally written in 1965, while Herbert Federer's massive treatise Geometric Measure Theory [F] was still in its early stages of prepara• tion. (That book was listed in the bibliography of the original edition of this book by name but with no other publication information— not even a publisher!) The notation and even the correct shape of the definitions in the field of geometric measure theory were still in a state of flux. In particular, one of the central objects defined by Fred, the no• tion of a /c-dimensional varifold, was later recast with a different def• inition by one of Fred's early collaborators, Bill Allard (who says Herbert Federer suggested it to him). The newer definition, a Radon measure on the product of R3 with the Grassmanian Qk of oriented (or unoriented) /c-planes through the origin in R3, is the one that became standard. In fact, Fred's original research paper on varifolds was never published, but exists only as a grey-covered monograph in places such as the Mathematics and Physics Library at Princeton University [A]. The definitive paper on varifolds became Bill Allard's On the first variation of a varifold [1]. Nevertheless, there is nothing wrong with the definition of varifold given initially and in this book. For those who know measure theory, it is not hard to translate the statements given in this book to the Radon measure formulation. (Specifically, if fi is a Radon measure as above, then the /c-dimensional varifold V defined in the notation of this book is given by v(</>)= f <p{P){P)d^P) 3 J(p,P)eR xQk for every differential A;-form <fi.) For those who do not, this book is self-contained in its exposition as is. There has been, of course, a great deal of soap-film mathematics created in the past thirty-odd years since the original publication date of this book. Fred gave here the ideas needed for his proof of the existence of area-minimizing surfaces on boundaries. Later, he proved regularity almost everywhere, not only for soap films and soap bubble clusters and bubbles in films, but also for the orientation-dependent Foreword to the AMS Edition XI surface energies that sometimes occur in crystals and other subjects of materials science [3]. I had the privilege to be Fred's first graduate student and, thereby, to get his best thesis problem, one about "flat chains modulo 3" that soon led to theorems about the points in soap films and bubbles which are not smooth portions of surface: I was able to prove that they con• sist of smooth curves along which three smooth pieces of surface meet at equal angles together with isolated point singularities where four of these triple junctions come together at equal angles [14]. This had been one of the fundamental observations of Plateau one hun• dred years earlier. Many of Fred's later graduate students proved additional theorems about soap bubbles and soap films, in particular Frank Morgan, Brian White, Ken Brakke, John Sullivan, and Sheldon Chang. Also, Ken Brakke has written and continues to update a mar• velous (and free!) computer program called the Surface Evolver [8] which, among its many virtuosities, computes soap bubbles and soap films. In particular, the 18 new illustrations for this edition (replacing earlier hand-drawn illustrations) were produced by Brakke and Sulli• van using the Evolver and Sullivan's rendering program.