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Transcript of Introduction to Geometric Measure Preface to the Tsinghua Lectures 2014 The present text is a...

  • Introduction to Geometric Measure Theory

    Leon Simon 1

    © Leon Simon 2014

    1 The research described here was partially supported by NSF grants DMS-9504456 & DMS–9207704 at Stanford University

  • Contents

    1 Preliminary Measure Theory 1

    1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Hausdorff Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Radon Measures, Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . 27

    2 Some Further Preliminaries from Analysis 39

    1 Lipschitz Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2 BV Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3 The Area Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4 Submanifolds of RnC` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5 First and Second Variation Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6 Co-Area Formula and C 1 Sard Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 65

    3 Countably n-Rectifiable Sets 69

    1 Basic Notions, Tangent Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2 Gradients, Jacobians, Area, Co-Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3 Purely Unrectifiable Sets, Structure Theorem . . . . . . . . . . . . . . . . . . . . . 78 4 Sets of Locally Finite Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

    4 Rectifiable n-Varifolds 85

    1 Basic Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2 First Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3 Monotonicity Formulae in the Stationary Case . . . . . . . . . . . . . . . . . . . 89 4 Monotonicity Formulae for Lp Mean Curvature . . . . . . . . . . . . . . . . . 93 5 Poincaré and Sobolev Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6 Miscellaneous Additional Consequences . . . . . . . . . . . . . . . . . . . . . . . . . 100

    5 The Allard Regularity Theorem 103

    1 Harmonic Approximation in the Smooth Case . . . . . . . . . . . . . . . . . . 104 2 Preliminaries, Lipschitz Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 106 3 Approximation by Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . 114 4 The Tilt-Excess Decay Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5 Main Regularity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6 Conical Approximation, Extension of Allard’s Th. . . . . . . . . . . . . . . 124 7 Some Initial Applications of the Allard Theorem . . . . . . . . . . . . . . . . 127

    6 Currents 131

    1 Preliminaries: Vectors, Co-vectors, and Forms . . . . . . . . . . . . . . . . . . . 131 2 General Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3 Integer Multiplicity Rectifiable Currents . . . . . . . . . . . . . . . . . . . . . . . . 145 4 Slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5 The Deformation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6 Applications of Deformation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7 The Flat Metric Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8 The Rectifiability and Compactness Theorems . . . . . . . . . . . . . . . . . . . 172

    7 Area Minimizing Currents 181

    1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 2 Existence and Compactness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 3 Tangent Cones and Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4 Some Regularity Results (Arbitrary Codimension) . . . . . . . . . . . . . . . 193 5 Codimension 1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

    8 Theory of General Varifolds 205

    1 Basics, First Rectifiability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 2 First Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 3 Monotonicity and Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 4 Constancy Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5 Varifold Tangents and Rectifiability Theorem . . . . . . . . . . . . . . . . . . . . 215

    A A General Regularity Theorem 225

    B Non-existence of Stable Minimal Cones, n � 6 229

  • Preface to the Tsinghua Lectures 2014

    The present text is a revision and updating of the author’s 1983 “Lectures on Ge- ometric Measure Theory,” and is meant to provide an introduction to the subject at beginning/intermediate graduate level. The present draft is still in rather rough form, with a generous scattering of (hopefully not serious, mainly expository) er- rors. During the Tsinghua lectures (February–April 2014) the notes will be further revised, with the ultimate aim of providing a useful and accessible introduction to the subject at the appropriate level.

    The author would greatly appreciate feedback about errors and other deficiencies.

    Leon Simon Beijing, China, February 2014 lsimon@stanford.edu

    Last updated: Wed Dec 3 at 3:05pm

    Notation SA D closure of A, assuming A is a subset of some topological space X

    B nA D {x 2 B W x … A} � A D indicator function of A (D 1 at points of A and D 0 and points not in A)

    IA D identity map A! A

    Ln D Lebesgue outer measure in Rn

    B�(y) D closed ball with center y radius � (more specifically denoted Bn� (y) if we wish to emphasize that we are working in Rn). Thus B�(y) D {x 2 Rn W jx � yj � �}, or more generally, in any metric space X , B�(y) D {x 2 X W d (x; y) � �}. MB�(y) D open ball D {x 2 Rn W jx � yj < �};

    !k D �k=2R 1

    0 tk=2e�t dt

    for k � 0 (so !k D Lk({x 2 Rk W jxj � 1}) if k 2 {1; 2; : : :}).

    �y;� W Rn ! Rn (for � > 0, y 2 Rn) is defined by �y;�(x) D ��1(x � y); thus �y;1 is translation x 7! x � y, and �0;� is homothety x 7! ��1x

    C k(U ;V ) (U , V open subsets of Euclidean spaces Rn and Rm respectively) denotes the space of C k maps from U into V

    For f 2 C 1(U ;V ), Df is the derivative matrix with entries Difj and the i -th row and j -th column, and jDf j2 D

    Pn iD1

    Pm jD1(Difj )

    2.

    For f 2 C 1(U ;R)withU open inRn, we writerf D (D1f; : : : ;Dnf ) (D (Df )T ).

    C kc (U ;V ) D {' 2 C k(U ;V ) W ' has compact support}

    For an abstract set X , 2X denotes the collection of all subsets of X

    ∅ D the empty set.

    For any set A in a metric space X with metric d , diamA denotes the diameter of the set A, i.e. supx;y2A d (x; y), interpreted to be zero if A is empty and 1 if A is not bounded.

    For � Rn open, W 1;2() will denote the Sobolev space of functions f W ! R such that f;rf 2 L2().

    If W � U , U open in Rn, W �� U means SW is a compact subset of U .

    ıij D Kronecker delta (D 1 if i D j , 0 if i ¤ j ).

  • Chapter 1

    Preliminary Measure Theory 1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Hausdorff Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Radon Measures, Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . 27

    In this chapter we briefly review the basic theory of outer measure, which is based on Caratheodory’s definition of measurability. Hausdorff (outer) measure is dis- cussed, including the main results concerning n-dimensional densities and the way in which they relate more general measures to Hausdorff measures. The final two sections of the chapter give the basic theory of Radon (outer) measures including the Riesz representation theorem and the standard differentiation theory for Radon measures.

    For the first section of the chapter X will denote an abstract space, and later we impose further restrictions on X as appropriate. For example in the second and third sections X is a metric space and in the last section of the chapter we shall assume that X is a locally compact, separable metric space.

    1 Basic Notions

    Recall that an outer measure (sometimes simply called a measure if no confusion is likely to arise) on X is a monotone subadditive function � W 2X ! [0;1] with �(∅) D 0. Thus �(∅) D 0, and

    1.1 �(A) � P1 jD1�(Aj ) whenever A � [

    1 jD1Aj

  • 2 Chapter 1: Preliminary Measure Theory

    with A, A1, A2, : : : any countable collection of subsets of X . Of course this in particular implies �(A) � �(B) whenever A