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 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 16 2-3. The Grassmann algebra and its dual 19 2-4. Differential forms 22 2-5. The Grassmann 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, 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 [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 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

JEAN E. TAYLOR March 2001 This page intentionally left blank Editors' Foreword

Mathematics has been expanding in all directions at a fabulous rate during the past half century. New fields have emerged, the diffu• sion into other disciplines has proceeded apace, and our knowledge of the classical areas has grown ever more profound. At the same time, one of the most striking trends in modern mathematics is the constantly increasing interrelationship between its various branches. Thus the present-day students of mathematics are faced with an im• mense mountain of material. In addition to the traditional areas of mathematics as presented in the traditional manner—and these presentations do abound—there are the new and often enlightening ways of looking at these traditional areas, and also the vast new areas teeming with potentialities. Much of this new material is scattered indigestibly throughout the research journals, and frequently coher• ently organized only in the minds or unpublished notes of the working mathematicians. And students desperately need to learn more and more of this material. This series1 of brief topical booklets has been conceived as a pos• sible means to tackle and hopefully to alleviate some of these peda• gogical problems. They are being written by active research math• ematicians, who can look at the latest developments, who can use these developments to clarify and condense the required material,

Publisher's note: W. A. Benjamin's Mathematics Monograph Series.

Xlll XIV Editors' Foreword who know what ideas to underscore and what techniques to stress. We hope that these books will also serve to present to the able un• dergraduate an introduction to contemporary research and problems in mathematics, and that they will be sufficiently informal that the personal tastes and attitudes of the leaders in modern mathematics will shine through clearly to the readers. Plateau's problem, roughly that of finding a surface of least area having a given boundary, has long challenged pure and applied math• ematicians by the beautiful simplicity of its statement but the sur• prising difficulties in the way of its solution. Attacks on this problem have prompted a good deal of work in the . Con• tinuing this direction of works, Dr. Almgren's book introduces the reader to a new and promising approach to the problem, reformulat• ing it in terms of currents and varifolds. A student with a course in advanced calculus behind him should find the book quite accessible, surprisingly so in view of the novelty of the material to the traditional undergraduate curriculum.

ROBERT GUNNING Princeton, New Jersey

HUGO ROSSI Waltham, Massachusetts

September 1965 Preface

Much has been written about the of three- dimensional . The properties of curves and surfaces have been studied in detail, utilizing the concepts of tangent vectors, normal vectors, curvatures, etc. It is not surprising, then, that one seeks to express mathematical problems about, say, two-dimensional surfaces in three-dimensional space in the language of differential ge• ometry. Indeed, until fairly recently, there has not been much alter• native. This book introduces a new language for the expression of many geometric concepts—the language of varifolds. Varifolds of dimensions one and two are curves and surfaces de• fined in Euclidean space in a measure theoretic way, and it has been only recently that enough has been understood about the relationship between measure theory and geometry to make possible the creation of these surfaces. The most important uses for varifolds have been in the calculus of variations, where the generality of varifolds permits results once im• possible to obtain. The peculiar advantage of varifolds is particularly well illustrated by considering a famous problem of modern mathe• matics, Plateau 7s problem, set in the context of varifolds. Plateau's problem is concerned with surfaces which have the geometric prop• erties of soap films (which are different from the properties of soap bubbles). For a given boundary curve one asks first whether or not

xv XVI Preface there exists any mathematical surface of this type at all, and, sec• ondly, whether or not there exists such a mathematical surface of smallest area. The initial difficulty of studying this problem in the usual language of differential geometry, assuming of course that one wishes to study surfaces which really resemble the soap films which form rather than an abstraction of the problem, is the difficulty of describing in that language exactly what a soap-filmlike surface is. In particular, the set of singularities which one has to include in the definition is not known. Even if it is possible some day to know all possible singularities, this would still leave the higher-dimensional ver• sions of the problem unsolved—and there are infinitely many higher dimensions! Perhaps the one most important virtue of varifolds is that it is possible to obtain a geometrically significant solution to a number of variational problems, including Plateau's problem, with• out having to know ahead of time what all the possible singularities of the solution can be. It is then important, of course, to determine what the singularities of solutions are, but this is a separate problem whose solution is not yet in sight. Although this book discusses explicitly only varifolds of dimen• sions zero, one, two, and three in three-dimensional space, virtually all the results and definitions are equally valid for /c-dimensional surfaces in n-dimensional space for all integers 0 < k < n. It is the purpose of this book to introduce the definitions of vari- fold geometry and to discuss the reasons for its creation. In the final chapter we state without proof of the chief results of the theory when applied to least area problems. The author is indebted to William P. Ziemer for reading the man• uscript and for many constructive suggestions and to William Prokos for rendering the illustrations. This book was written while the au• thor was a member of the Institute for Advanced Study in Princeton, New Jersey, supported by grant GP2439 from the National Science Foundation.

FREDERICK J. ALMGREN, JR. Princeton, New Jersey September 1965 References

F. J. Almgren, Jr., The theory of varifolds—A variational calculus in the large for the k-dimensional area integrated. C. V. Boys, Soap Bubbles: Their Colours and the Forces Which Mold Them (New York: Dover Press, 1959). J. Douglas, "Minimal surfaces of higher topological structure," Annals of Mathematics 40, 205-298 (1939). H. Federer, "The (

73 74 References

E. R. Reifenberg, "An epiperimetric inequality related to the analyticity of minimal surfaces," Annals of Mathematics (1) 80, 1-14 (1964). E. R. Reifenberg, "On the analyticity of minimal surfaces," Annals of Mathematics (1) 80, 15-21 (1964). H. Whitney, Geometric Integration Theory (Princeton, N.J.: Princeton University Press, 1957). Additional References

[1] William K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417-491. [2] Frederick J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of vary• ing topological type and singularity structure, Ann. of Math. (2) 87 (1968), 321-391. [3] Frederick J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), no. 165. [4] Frederick J. Almgren, Jr., Selected Works, American Mathematical Society, Providence, RI, 1999. [5] Frederick J. Almgren, Jr., Almgren's Big Regularity Paper: Q-Valued Functions Minimizing DirichleVs Integral and the Regularity of Area- Minimizing Rectifiable Currents up to Codimension 2, World Scientific Publishing Co., Singapore, 2000. [6] Frederick J. Almgren, Jr., and Jean E. Taylor, The geometry of soap films, Scientific American, July (1976), 82-93. [7] Frederick J. Almgren, Jr., and Jean E. Taylor, Soap bubble clusters, Forma 11 (1996), 199-207. [8] Kenneth A. Brakke, The surface evolver, Experiment. Math. 1 (1992), no. 2, 141-165. [9] Kenneth A. Brakke, Soap films and covering spaces, J. Geom. Anal. 5 (1995), no. 4, 445-514. [10] Special Issue dedicated to the memory of Frederick J. Almgren, Jr., J. Geom. Anal, 8, (1998) no. 5.

75 76 Additional References

[11] Dana Mackenzie, Fred Almgren (1933-1997): Lover of mathematics, family, and life's adventures, Notices Amer. Math. Soc, October, 1997, 1102-1106. [12] Frank Morgan, Geometric Measure Theory. A beginner's guide. Third edition. Academic Press, Inc., San Diego, CA, 2000. [13] Frank Morgan, Proof of the double bubble conjecture, Amer. Math. Monthly, March, 2001. [14] Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces. Ann. of Math. (2) 103 (1976), no. 3, 489-539. [15] Jean E. Taylor, The structure of singularities in solutions to ellipsoidal variational problems with constraints in R3. Ann. of Math. (2) 103 (1976), no. 3, 541-546. [16] Brian White, The mathematics of F. J. Almgren, Jr., J. Geom. Anal, 8 (1998), no. 5, 681-702. A shorter version appears in Notices Amer. Math. Soc, December, 1997, 1451-1456. [17] Laurence C. Young, Generalized surfaces in the calculus of variations I, II, Ann. of Math. 43 (1942), 84-103, 530-544. Index

Analytic sets, 19 Lebesgue measure, 19 Area, 17-19 Length, 17, 19 Lipschitz condition, 15-22 C\ 22 Calculus of variations, 56 M, 47 Covectors, 19-22 Mass, 51 Critical point, 56 Metric, 23 Current, 39 distance, 23 integral, 4, 40 see Varifold metrics rectifiable, 39 Minimal surface, 65 Mobius band, 5-8 triple, 6-8 Diameter of regular varifold, 72 Monotonicity for regular varifolds, Disk, 3 70, 71 with handles, 3 Multiplicities, 2, 40, 41, 48 with spines, 42-43 Distance, see Metric Douglas, J., 2, 3, 73 P, 61 Dual vector space, 19-22 Partial ordering of varifolds, see Varifold operations Plateau, J., 1, 73 F, 45 Plateau's problem, 1 Federer, H., 4, 68, 73 construction problem, 2 Flat chains with coefficients in a existence problem, 2 group, 5 Projective space, 23 Fleming, W. H., 4, 69, 73 Force diagram method, 64 , 31-34 oriented, 34

GVfc, 53 Reifenberg, E. R., 4, 69, 73

S, 57 Hfc, 17 Singular set, 13 Soap bubbles, 1-2, 58, 61, 66, 67 Intersection of varifolds, see Varifold Soap films, 1-2, 10-13, 58-61 operations not touching all of boundary wire, Isoperimetric inequalities, 66, 72 8-9, 66, 69, 70 IVfc, 44 Stationary point, 56

77 78 Index

Stationary varifold, see Varifold Stokes' theorem, 39 Sum of varifolds, see Varifold operations Support, of , 22 of varifold, 44 Surface tension, 10 Suslin sets, 19

T, 59

Union of varifolds, see Varifold operations v, 52 Vfc, 44 Variation, 57 Varifold, axioms for, 38, 44 definition, 44 elementary, 52-53 geometric, 52 integral, 1, 39, 47 rectifiable, 39 stationary, 56, 59 Varifold metrics, 45, 47 Varifold operations intersection, 44 partial ordering, 44 sum, 44 union, 44 Varifold pairs, 61 regular, 61 stationary, 61 touching all of boundary, 69

W, 48

X, 55