DOCSLIB.ORG

View This Volume's Front and Back Matter

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
View This Volume's Front and Back Matter 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.
Load more

Recommended publications
  • The Singular Structure and Regularity of Stationary Varifolds 3
    THE SINGULAR STRUCTURE AND REGULARITY OF STATIONARY VARIFOLDS AARON NABER AND DANIELE VALTORTA ABSTRACT. If one considers an integral varifold Im M with bounded mean curvature, and if S k(I) x ⊆ ≡ { ∈ M : no tangent cone at x is k + 1-symmetric is the standard stratification of the singular set, then it is well } known that dim S k k. In complete generality nothing else is known about the singular sets S k(I). In this ≤ paper we prove for a general integral varifold with bounded mean curvature, in particular a stationary varifold, that every stratum S k(I) is k-rectifiable. In fact, we prove for k-a.e. point x S k that there exists a unique ∈ k-plane Vk such that every tangent cone at x is of the form V C for some cone C. n 1 n × In the case of minimizing hypersurfaces I − M we can go further. Indeed, we can show that the singular ⊆ set S (I), which is known to satisfy dim S (I) n 8, is in fact n 8 rectifiable with uniformly finite n 8 measure. ≤ − − − 7 An effective version of this allows us to prove that the second fundamental form A has apriori estimates in Lweak on I, an estimate which is sharp as A is not in L7 for the Simons cone. In fact, we prove the much stronger | | 7 estimate that the regularity scale rI has Lweak-estimates. The above results are in fact just applications of a new class of estimates we prove on the quantitative k k k k + stratifications S ǫ,r and S ǫ S ǫ,0.
    [Show full text]
  • Geometric Measure Theory and Differential Inclusions
    GEOMETRIC MEASURE THEORY AND DIFFERENTIAL INCLUSIONS C. DE LELLIS, G. DE PHILIPPIS, B. KIRCHHEIM, AND R. TIONE Abstract. In this paper we consider Lipschitz graphs of functions which are stationary points of strictly polyconvex energies. Such graphs can be thought as integral currents, resp. varifolds, which are stationary for some elliptic integrands. The regularity theory for the latter is a widely open problem, in particular no counterpart of the classical Allard’s theorem is known. We address the issue from the point of view of differential inclusions and we show that the relevant ones do not contain the class of laminates which are used in [22] and [25] to construct nonregular solutions. Our result is thus an indication that an Allard’s type result might be valid for general elliptic integrands. We conclude the paper by listing a series of open questions concerning the regularity of stationary points for elliptic integrands. 1. Introduction m 1 n×m Let Ω ⊂ R be open and f 2 C (R ; R) be a (strictly) polyconvex function, i.e. such that there is a (strictly) convex g 2 C1 such that f(X) = g(Φ(X)), where Φ(X) denotes the vector of n subdeterminants of X of all orders. We then consider the following energy E : Lip(Ω; R ) ! R: E(u) := f(Du)dx : (1) ˆΩ n Definition 1.1. Consider a map u¯ 2 Lip(Ω; R ). The one-parameter family of functions u¯ + "v will be called outer variations and u¯ will be called critical for E if d E 1 n (¯u + "v) = 0 8v 2 Cc (Ω; R ) : d" "=0 1 m 1 Given a vector field Φ 2 Cc (Ω; R ) we let X" be its flow .
    [Show full text]
  • GMT – Varifolds Cheat-Sheet Sławomir Kolasiński Some Notation
    GMT – Varifolds Cheat-sheet Sławomir Kolasiński Some notation [id & cf] The identity map on X and the characteristic function of some E ⊆ X shall be denoted by idX and 1E : [Df & grad f] Let X, Y be Banach spaces and U ⊆ X be open. For the space of k times continuously k differentiable functions f ∶ U → Y we write C (U; Y ). The differential of f at x ∈ U is denoted Df(x) ∈ Hom(X; Y ) : In case Y = R and X is a Hilbert space, we also define the gradient of f at x ∈ U by ∗ grad f(x) = Df(x) 1 ∈ X: [Fed69, 2.10.9] Let f ∶ X → Y . For y ∈ Y we define the multiplicity −1 N(f; y) = cardinality(f {y}) : [Fed69, 4.2.8] Whenever X is a vectorspace and r ∈ R we define the homothety µr(x) = rx for x ∈ X: [Fed69, 2.7.16] Whenever X is a vectorspace and a ∈ X we define the translation τ a(x) = x + a for x ∈ X: [Fed69, 2.5.13,14] Let X be a locally compact Hausdorff space. The space of all continuous real valued functions on X with compact support equipped with the supremum norm is denoted K (X) : [Fed69, 4.1.1] Let X, Y be Banach spaces, dim X < ∞, and U ⊆ X be open. The space of all smooth (infinitely differentiable) functions f ∶ U → Y is denoted E (U; Y ) : The space of all smooth functions f ∶ U → Y with compact support is denoted D(U; Y ) : It is endowed with a locally convex topology as described in [Men16, Definition 2.13].
    [Show full text]
  • GMT Seminar: Introduction to Integral Varifolds
    GMT Seminar: Introduction to Integral Varifolds DANIEL WESER These notes are from two talks given in a GMT reading seminar at UT Austin on February 27th and March 6th, 2019. 1 Introduction and preliminary results Definition 1.1. [Sim84] n k Let G(n; k) be the set of k-dimensional linear subspaces of R . Let M be locally H -rectifiable, n 1 and let θ : R ! N be in Lloc. Then, an integral varifold V of dimension k in U is a Radon 0 measure on U × G(n; k) acting on functions ' 2 Cc (U × G(n; k) by Z k V (') = '(x; TxM) θ(x) dH : M By \projecting" U × G(n; k) onto the first factor, we arrive at the following definition: Definition 1.2. [Lel12] n Let U ⊂ R be an open set. An integral varifold V of dimension k in U is a pair V = (Γ; f), k 1 where (1) Γ ⊂ U is a H -rectifiable set, and (2) f :Γ ! N n f0g is an Lloc Borel function (called the multiplicity function of V ). We can naturally associate to V the following Radon measure: Z k µV (A) = f dH for any Borel set A: Γ\A We define the mass of V to be M(V ) := µV (U). We define the tangent space TxV to be the approximate tangent space of the measure µV , k whenever this exists. Thus, TxV = TxΓ H -a.e. Definition 1.3. [Lel12] If Φ : U ! W is a diffeomorphism and V = (Γ; f) an integral varifold in U, then the pushfor- −1 ward of V is Φ#V = (Φ(Γ); f ◦ Φ ); which is itself an integral varifold in W .
    [Show full text]
  • Intrinsic Geometry of Varifolds in Riemannian Manifolds: Monotonicity and Poincare-Sobolev Inequalities
    Intrinsic Geometry of Varifolds in Riemannian Manifolds: Monotonicity and Poincare-Sobolev Inequalities Julio Cesar Correa Hoyos Tese apresentada ao Instituto de Matemática e Estatística da Universidade de São Paulo para obtenção do título de Doutor em Ciências Programa: Doutorado em Matemática Orientador: Prof. Dr. Stefano Nardulli Durante o desenvolvimento deste trabalho o autor recebeu auxílio financeiro da CAPES e CNPq São Paulo, Agosto de 2020 Intrinsic Geometry of Varifolds in Riemannian Manifolds: Monotonicity and Poincare-Sobolev Inequalities Esta é a versão original da tese elaborada pelo candidato Julio Cesar Correa Hoyos, tal como submetida à Comissão Julgadora. Resumo CORREA HOYOS, J.C. Geometría Intrínsica de Varifolds em Variedades Riemannianas: Monotonia e Desigualdades do Tipo Poincaré-Sobolev . 2010. 120 f. Tese (Doutorado) - Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 2020. São provadas desigualdades do tipo Poincaré e Sobolev para funções com suporte compacto definidas em uma varifold k-rectificavel V definida em uma variedade Riemanniana com raio de injetividade positivo e curvatura secional limitada por cima. As técnica usadas permitem consid- erar variedades Riemannianas (M n; g) com métrica g de classe C2 ou mais regular, evitando o uso do mergulho isométrico de Nash. Dito análise permite re-fazer algums fragmentos importantes da teoría geométrica da medida no caso das variedades Riemannianas com métrica C2, que não é Ck+α, com k + α > 2. A classe de varifolds consideradas, são aquelas em que sua primeira variação δV p esta em um espaço de Labesgue L com respeito à sua medida de massa kV k com expoente p 2 R satisfazendo p > k.
    [Show full text]
  • Rapid Course on Geometric Measure Theory and the Theory of Varifolds
    Crash course on GMT Sławomir Kolasiński Rapid course on geometric measure theory and the theory of varifolds I will present an excerpt of classical results in geometric measure theory and the theory of vari- folds as developed in [Fed69] and [All72]. Alternate sources for fragments of the theory are [Sim83], [EG92], [KP08], [KP99], [Zie89], [AFP00]. Rectifiable sets and the area and coarea formulas (1) Area and coarea for maps between Euclidean spaces; see [Fed69, 3.2.3, 3.2.10] or [KP08, 5.1, 5.2] (2) Area and coarea formula for maps between rectifiable sets; see [Fed69, 3.2.20, 3.2.22] or [KP08, 5.4.7, 5.4.8] (3) Approximate tangent and normal vectors and approximate differentiation with respect to a measure and dimension; see [Fed69, 3.2.16] (4) Approximate tangents and densities for Hausdorff measure restricted to an (H m; m) recti- fiable set; see [Fed69, 3.2.19] (5) Characterization of countably (H m; m)-rectifiable sets in terms of covering by submanifolds of class C 1; see [Fed69, 3.2.29] or [KP08, 5.4.3] Sets of finite perimeter (1) Gauss-Green theorem for sets of locally finite perimeter; see [Fed69, 4.5.6] or [EG92, 5.7–5.8] (2) Coarea formula for real valued functions of locally bounded variation; see [Fed69, 4.5.9(12)(13)] or [EG92, 5.5 Theorem 1] (3) Differentiability of approximate and integral type for real valued functions of locally bounded variation; see [Fed69, 4.5.9(26)] or [EG92, 6.1 Theorems 1 and 4] (4) Characterization of sets of locally finite perimeter in terms of the Hausdorff measure of the measure theoretic boundary; see [Fed69, 4.5.11] or [EG92, 5.11 Theorem 1] Varifolds (1) Notation and basic facts for submanifolds of Euclidean space, see [All72, 2.5] or [Sim83, §7] (2) Definitions and notation for varifolds, push forward by a smooth map, varifold disintegration; see [All72, 3.1–3.3] or [Sim83, pp.
    [Show full text]
  • The Width of Ellipsoids
    The width of Ellipsoids Nicolau Sarquis Aiex February 29, 2016 Abstract. We compute the ‘k-width of a round 2-sphere for k = 1,..., 8 and we use this result to show that unstable embedded closed geodesics can arise with multiplicity as a min-max critical varifold. 1 Introduction The aim of this work is to compute some of the k-width of the 2-sphere. Even in this simple case the full width spectrum is not very well known. One of the motivations is to prove a Weyl type law for the width as it was proposed in [9], where the author suggests that the width should be considered as a non-linear spectrum analogue to the spectrum of the laplacian. In a closed Riemannian manifold M of dimension n the Weyl law says that 2 λp − th 2 → cnvol(M) n for a known constant cn, where λp denotes the p eigenvalue p n of the laplacian. In the case of curves in a 2-dimensional manifold M we expect that ω 1 p − 2 1 → C2vol(M) , p 2 where C2 > 0 is some constant to be determined. We were unable to compute all of the width of S2 but we propose a general formula that is consistent with our results and the desired Weyl law. In the last section we explain it in more details. By making a contrast with classical Morse theory one could ask the following arXiv:1601.01032v2 [math.DG] 26 Feb 2016 two naive questions about the index and nullity of a varifold that achieves the width: Question 1: Let (M,g) be a Riemannian manifold and V ∈ IVl(M) be a critical varifold for the k-width ωk(M,g).
    [Show full text]
  • An Introduction to Varifolds
    AN INTRODUCTION TO VARIFOLDS DANIEL WESER These notes are from a talk given in the Junior Analysis seminar at UT Austin on April 27th, 2018. 1 Introduction Why varifolds? A varifold is a measure-theoretic generalization of a differentiable submanifold of Euclidean space, where differentiability has been replaced by rectifiability. Varifolds can model almost any surface, including those with singularities, and they've been used in the following applications: 1. for Plateau's problem (finding area-minimizing soap bubble-like surfaces with boundary values given along a curve), varifolds can be used to show the existence of a minimal surface with the given boundary, 2. varifolds were used to show the existence of a generalized minimal surface in given compact smooth Riemannian manifolds, 3. varifolds were the starting point of mean curvature flow, after they were used to create a mathematical model of the motion of grain boundaries in annealing pure metals. 2 Preliminaries We begin by covering the needed background from geometric measure theory. Definition 2.1. [Mag12] n Given k 2 N, δ > 0, and E ⊂ R , the k-dimensional Hausdorff measure of step δ is k k X diam(F ) Hδ = inf !k ; F 2 F 2F n where F is a covering of E by sets F ⊂ R such that diam(F ) < δ. Definition 2.2. [Mag12] The k-dimensional Hausdorff measure of E is k lim Hδ (E): δ!0+ 1 Definition 2.3. [Mag12] n k k Given a set M ⊂ R , we say that M is k-rectifiable if (1) M is H measurable and H (M) < k n +1, and (2) there exists countably many Lipschitz maps fj : R ! R such that k [ k H M n fj(R ) = 0: j2N Remark.
    [Show full text]
  • Perspectives in Geometric Analysis-- Joint Event of ANU-BICMR-PIMS-UW
    Perspectives in Geometric Analysis-- Joint event of ANU-BICMR-PIMS-UW Beijing International Center for Mathematical Research Lecture Hall, Jia Yi Bing Building, 82# Jing Chun Yuan Schedule and abstracts Schedule: 6.27 Monday morning 8:30-9:30 Xiaodong Wang, Comparison and rigidity results on compact Riemannian manifolds with boundary 9:40-10:40 Richard Bamler, Ricci flows with bounded scalar curvature 11:00-12:00 Jeff Streets, Generalized Kahler-Ricci flow Monday afternoon 2:00-3:00 Rivière Tristan, The Variations of Yang-Mills Lagrangian 3:10-4:10 Jiazu Zhou, Isoperimetric problems and Alexandrov-Fenchel inequalities 6.28 Tuesday morning 8:30-9:30 Xiaodong Wang, Comparison and rigidity results on compact Riemannian manifolds with boundary 9:40-10:40 Richard Bamler, Ricci flows with bounded scalar curvature 11:00-12:00 Jeff Streets, Generalized Kahler-Ricci flow Tuesday afternoon 2:00-3:00 Rivière Tristan, The Variations of Yang-Mills Lagrangian 3:10-4:10 Claudio Arezzo, Kahler constant scalar curvature metrics on blow ups and resolutions of singularities 4:30-5:30 Haotian Wu, Asymptotic shapes of neckpinch singularities in geometric flows 6.29 Wednesday morning 8:30-9:30 Xiaodong Wang, Comparison and rigidity results on compact Riemannian manifolds with boundary 9:40-10:40 Richard Bamler, Ricci flows with bounded scalar curvature 11:00-12:00 Jeff Streets, Generalized Kahler-Ricci flow Wednesday afternoon 2:00-3:00 Xianzhe Dai, Index theory and its geometric applications 3:10-4:10 Xianzhe Dai, Index theory and its geometric applications
    [Show full text]
  • Max-Planck-Institut Für Mathematik in Den Naturwissenschaften Leipzig
    Max-Planck-Institut fur¨ Mathematik in den Naturwissenschaften Leipzig Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies by Guido De Philippis, Antonio De Rosa, and Francesco Ghiraldin Preprint no.: 61 2016 RECTIFIABILITY OF VARIFOLDS WITH LOCALLY BOUNDED FIRST VARIATION WITH RESPECT TO ANISOTROPIC SURFACE ENERGIES GUIDO DE PHILIPPIS, ANTONIO DE ROSA, AND FRANCESCO GHIRALDIN Abstract. We extend Allard's celebrated rectifiability theorem to the setting of varifolds with locally bounded first variation with respect to an anisotropic integrand. In particular, we identify a sufficient and necessary condition on the integrand to obtain the rectifiability of every d-dimensional varifold with locally bounded first variation and positive d-dimensional density. In codimension one, this condition is shown to be equivalent to the strict convexity of the integrand with respect to the tangent plane. 1. Introduction n Allard's rectifiability Theorem, [1], asserts that every d-varifold in R with locally bounded (isotropic) first variation is d-rectifiable when restricted to the set of points n in R with positive lower d-dimensional density. It is a natural question whether this result holds for varifolds whose first variation with respect to an anisotropic integrand is locally bounded. n 1 More specifically, for an open set Ω ⊂ R and a positive C integrand F :Ω × G(n; d) ! R>0 := (0; +1); n where G(n; d) denotes the Grassmannian of d-planes in R , we define the anisotropic energy of a d-varifold V 2 Vd(Ω) as Z F(V; Ω) := F (x; T ) dV (x; T ): (1.1) Ω×G(n;d) We also define its anisotropic first variation as the order one distribution whose 1 n action on g 2 Cc (Ω; R ) is given by d # δF V (g) := F 't V dt t=0 Z h i = hdxF (x; T ); g(x)i + BF (x; T ): Dg(x) dV (x; T ); Ω×G(n;d) # n where 't(x) = x+tg(x), 't V is the image varifold of V through 't, BF (x; T ) 2 R ⊗ n ∗ n n R is an explicitly computable n×n matrix and hA; Bi := tr A B for A; B 2 R ⊗R , see Section 2 below for the precise definitions and the relevant computations.
    [Show full text]
  • How Riemannian Manifolds Converge: a Survey
    How Riemannian Manifolds Converge: A Survey Christina Sormani, CUNY 1 ABSTRACT: This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean space: Hausdorff convergence of sets, flat con- vergence of integral currents, and weak convergence of varifolds. We next describe a variety of intrinsic notions of convergence which have been applied to study sequences of compact Riemann- ian manifolds: Gromov-Hausdorff convergence of metric spaces, convergence of metric measure spaces, Instrinsic Flat convergence of integral current spaces, and ultralimits of metric spaces. We close with a speculative section addressing possible notions of intrinsic varifold convergence, convergence of Lorentzian manifolds and area convergence. 1. Introduction The strong notions of smoothly or Lipschitz converging manifolds have proven to be excep- tionally useful when studying manifolds with curvature and volume bounds, Einstein manifolds, isospectral manifolds of low dimensions, conformally equivalent manifolds, Ricci flow and the Poincare conjecture, and even some questions in general relativity. However many open ques- tions require weaker forms of convergence that do not produce limit spaces that are manifolds themselves. Weaker notions of convergence and new notions of limits have proven necessary in the study of manifolds with no curvature bounds or only lower bounds on Ricci or scalar cur- vature, isospectral manifolds of higher dimension, Ricci flow through singularities, and general relativity. Here we survey a variety of weaker notions of convergence and the corresponding limit spaces covering both well established concepts, newly discovered ones and speculations.
    [Show full text]
  • Existence of Integral M-Varifolds Minimizing
    Existence of Integral m-Varifolds minimizing |A|p and |H|p, p>m, in Riemannian Manifolds R R Andrea MONDINO1 abstract. We prove existence of integral rectifiable m-dimensional varifolds minimizing functionals of the type |H|p and |A|p in a given Riemannian n-dimensional manifold (N,g), 2 ≤ m<n and p>m, under suitable assumptions on N (in the end of the paper we give many examples of such ambient manifolds).R ToR this aim we introduce the following new tools: some monotonicity formulas for varifolds in RS involving |H|p, to avoid degeneracy of the minimizer, and a sort of isoperimetric inequality to bound the mass in terms of the mentioned functionals. R Key Words: Curvature varifolds, direct methods in the calculus of variation, geometric measure theory, isoperimetric inequality, monotonicity formula AMS subject classification: 49Q20, 58E99, 53A10, 49Q05. 1 Introduction Given an ambient Riemannian manifold (N,g) of dimension n ≥ 3 (with or without boundary), a classical problem in differential geometry is to find the smooth immersed m-dimensional submanifolds, 2 ≤ m ≤ n − 1, with null mean curvature vector, H = 0, or with null second fundamental form, A = 0, namely the minimal (respectively, the totally geodesic) submanifolds of N (for more details about the existence see Example 7.3, Example 7.4, Theorem 7.6, Theorem 7.7, Remark 7.8 and Remark 7.9). In more generality, it is interesting to study the minimization problems associated to integral func- tionals depending on the curvatures of the type arXiv:1010.4514v2 [math.DG] 24 Jan 2014 p p p p (1) EH,m(M) := |H| or EA,m(M) := |A| , p ≥ 1 ZM ZM where M is a smooth immersed m-dimensional submanifold with mean curvature H and second funda- mental form A; of course the integrals are computed with respect to the m-dimensional measure of N p p induced on M.
    [Show full text]