Curvature measures of singular sets

Preface......Page 6
Contents......Page 9
  1.1.1 Hausdorff Measure......Page 12
  1.1.2 Densities of Sets......Page 14
  1.1.3 Lipschitz Mappings......Page 15
  1.1.5 Approximate Differential......Page 18
  1.1.6 Rectifiable Sets......Page 19
  1.1.7 Purely Unrectifiable Sets......Page 21
  1.2.1 Multilinear Algebra......Page 22
  1.2.2 The Grassmannian......Page 26
  1.2.3 Multivector Fields and Differential Forms......Page 28
  1.3.1 Basic Notions......Page 30
  1.3.2 Normal and Rectifiable Currents......Page 32
  1.3.3 Product of Currents......Page 33
  1.3.4 Push-Forward......Page 34
  1.3.5 Constancy Theorems......Page 36
  1.3.6 Integral Currents and Compactness......Page 39
  1.3.7 Slicing of Currents......Page 40
  1.3.8 Indecomposable Integral Currents......Page 41
 1.4 Computations on Grassmannians......Page 42
 2.1 Intrinsic Volumes......Page 48
 2.2 Curvature and Area Measures......Page 50
 2.3 Integral-Geometric Formulas......Page 52
 2.5 Simplicial Complexes......Page 53
 3.1 Curvature Measures of Smooth Submanifolds......Page 57
 3.2 Euler Characteristic......Page 60
 3.4 Morse Theory......Page 61
 4.1 Geometric Properties......Page 64
 4.2 Reach of Intersection......Page 72
 4.3 Unit Normal Bundle......Page 75
 4.4 Principal Curvatures and Directions......Page 77
 4.5 Curvature Measures and Steiner Formula......Page 80
 4.6 Normal Cycle......Page 86
 4.7 Basic Properties of Curvature Measures......Page 89
 4.8 Gauss-Bonnet Formula......Page 92
 4.9 Bibliographical Notes......Page 94
 5.1 Topological Index Functions and Additive Extension of Normal Cycles......Page 96
 5.2 Curvature Measures, Generalized Principal Curvatures and Steiner Formula......Page 102
 5.3 Regular UPR-Representations and the Reflection Principle for Normal Cycles......Page 106
 5.4 Bibliographical Notes......Page 111
6 Integral Geometric Formulas......Page 113
 6.1 Translative and Kinematic Integral Formulas for Curvature Measures......Page 114
 6.2 Local Representation of Mixed Curvature Measures......Page 126
 6.3 Absolute Curvature Measures......Page 136
 6.4 Bibliographical Notes......Page 144
 7.1 Approximation by Parallel Sets......Page 147
 7.2 Polytopal Approximation......Page 156
 7.3 Bibliographical Notes......Page 165
 8.1 Characterization of Lipschitz-Killing Curvatures......Page 167
 8.2 Characterization of Associated Measures......Page 172
 8.3 Bibliographical Notes......Page 177
9 Extensions of Curvature Measures to Larger Set Classes......Page 179
 9.1 Legendrian Cycles......Page 180
 9.2 Normal Cycles......Page 187
 9.3 Lipschitz Domains......Page 193
 9.4 Bibliographical Notes......Page 214
 10.1 Introduction......Page 216
 10.2 Self-Similar Sets......Page 224
  10.3.1 Characterization in Terms of Surface Area......Page 230
  10.3.2 Densities and Measures via Ergodic Theory......Page 231
  10.3.3 Total Values, Minkowski Measures and the Renewal Theorem......Page 233
  10.3.4 Relationships to Fractal Zeta Functions......Page 241
  10.4.1 Application of Renewal Theory......Page 242
  10.4.2 Methods from Ergodic Theory: Curvature Densities and Measures......Page 246
References......Page 252
List of Symbols......Page 257
Index......Page 260