The Singular Structure and Regularity of Stationary Varifolds 3
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Geometric Integration Theory Contents
Steven G. Krantz Harold R. Parks Geometric Integration Theory Contents Preface v 1 Basics 1 1.1 Smooth Functions . 1 1.2Measures.............................. 6 1.2.1 Lebesgue Measure . 11 1.3Integration............................. 14 1.3.1 Measurable Functions . 14 1.3.2 The Integral . 17 1.3.3 Lebesgue Spaces . 23 1.3.4 Product Measures and the Fubini–Tonelli Theorem . 25 1.4 The Exterior Algebra . 27 1.5 The Hausdorff Distance and Steiner Symmetrization . 30 1.6 Borel and Suslin Sets . 41 2 Carath´eodory’s Construction and Lower-Dimensional Mea- sures 53 2.1 The Basic Definition . 53 2.1.1 Hausdorff Measure and Spherical Measure . 55 2.1.2 A Measure Based on Parallelepipeds . 57 2.1.3 Projections and Convexity . 57 2.1.4 Other Geometric Measures . 59 2.1.5 Summary . 61 2.2 The Densities of a Measure . 64 2.3 A One-Dimensional Example . 66 2.4 Carath´eodory’s Construction and Mappings . 67 2.5 The Concept of Hausdorff Dimension . 70 2.6 Some Cantor Set Examples . 73 i ii CONTENTS 2.6.1 Basic Examples . 73 2.6.2 Some Generalized Cantor Sets . 76 2.6.3 Cantor Sets in Higher Dimensions . 78 3 Invariant Measures and the Construction of Haar Measure 81 3.1 The Fundamental Theorem . 82 3.2 Haar Measure for the Orthogonal Group and the Grassmanian 90 3.2.1 Remarks on the Manifold Structure of G(N,M).... 94 4 Covering Theorems and the Differentiation of Integrals 97 4.1 Wiener’s Covering Lemma and its Variants . -
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 . -
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]. -
Notes on Rectifiability
Notes on Rectifiability Urs Lang 2007 Version of September 1, 2016 Abstract These are the notes to a first part of a lecture course on Geometric Measure Theory I taught at ETH Zurich in Fall 2006. They partially overlap with [Lan1]. Contents 1 Embeddings of metric spaces 2 2 Compactness theorems for metric spaces 3 3 Lipschitz maps 5 4 Differentiability of Lipschitz maps 8 5 Extension of smooth functions 11 6 Metric differentiability 11 7 Hausdorff measures 15 8 Area formula 17 9 Rectifiable sets 20 10 Coarea formula 22 References 29 1 1 Embeddings of metric spaces We start with some basic and well-known isometric embedding theorems for metric spaces. 1 1 For a non-empty set X, l (X) = (l (X); k · k1) denotes the Banach space of all functions s: X ! R with ksk1 := supx2X js(x)j < 1. Similarly, 1 1 l := l (N) = f(sk)k2N : k(sk)k1 := supk2N jskj < 1g. 1.1 Proposition (Kuratowski, Fr´echet) (1) Every metric space X admits an isometric embedding into l1(X). (2) Every separable metric space admits an isometric embedding into l1. Proof : (1) Fix a basepoint z 2 X and define f : X ! l1(X), x 7! sx; sx(y) = d(x; y) − d(y; z): x x Note that ks k1 = supy js (y)j ≤ d(x; z). Moreover, x x0 0 0 ks − s k1 = supyjd(x; y) − d(x ; y)j ≤ d(x; x ); and equality occurs for y = x0. (2) Choose a basepoint z 2 X and a countable dense set D = fxk : k 2 1 Ng in X. -
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 . -
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. -
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. -
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). -
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. -
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 -
Anisotropic Energies in Geometric Measure Theory
A NISOTROPIC E NERGIESIN G EOMETRIC M EASURE T HEORY Dissertation zur Erlangung der naturwissenschaftlichen Doktorwürde (Dr. sc. nat.) vorgelegt der Mathematisch-naturwissenschaftlichen Fakultät der Universität Zürich von antonio de rosa aus Italien Promotionskommission Prof. Dr. Camillo De Lellis (Vorsitz) Prof. Dr. Guido De Philippis Prof. Dr. Thomas Kappeler Prof. Dr. Benjamin Schlein Zürich, 2017 ABSTRACT In this thesis we focus on different problems in the Calculus of Variations and Geometric Measure Theory, with the common peculiarity of dealing with anisotropic energies. We can group them in two big topics: 1. The anisotropic Plateau problem: Recently in [37], De Lellis, Maggi and Ghiraldin have proposed a direct approach to the isotropic Plateau problem in codimension one, based on the “elementary” theory of Radon measures and on a deep result of Preiss concerning rectifiable measures. In the joint works [44],[38],[43] we extend the results of [37] respectively to any codimension, to the anisotropic setting in codimension one and to the anisotropic setting in any codimension. For the latter result, we exploit the anisotropic counterpart of Allard’s rectifiability Theorem, [2], which we prove in [42]. It asserts that every d-varifold in Rn with locally bounded anisotropic first variation is d-rectifiable when restricted to the set of points in Rn with positive lower d-dimensional density. In particular we identify a necessary and sufficient condition on the Lagrangian for the validity of the Allard type rectifiability result. We are also able to prove that in codimension one this condition is equivalent to the strict convexity of the integrand with respect to the tangent plane. -
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.