Asymptotic Formulae in Spectral Geometry

Asymptotic Formulae in Spectral Geometry......Page 2
Studies in Advanced Mathematics......Page 3
Preface......Page 6
Contents......Page 9
Introduction......Page 11
Table of Contents......Page 0
The interior geometry......Page 13
The boundary geometry......Page 15
Covariant Differentiation......Page 17
Clifford Algebras......Page 18
Cli ord bundles......Page 21
Duality......Page 22
Volume of spheres......Page 23
The symbol of an operator......Page 24
An invariant representation of an operator of Laplace type......Page 25
The Dual Operator......Page 26
The form valued Laplacian......Page 27
The Hodge  operator......Page 29
The Witten Laplacian......Page 30
Singular Structures......Page 31
Interior ellipticity......Page 32
Elliptic regularity......Page 33
Heat trace asymptotics......Page 34
The Mellin transform......Page 36
Index Theory......Page 38
Heat content asymptotics......Page 40
Boundary ellipticity......Page 41
The heat equation......Page 43
Heat trace asymptotics......Page 44
Heat content asymptotics......Page 45
Operators of Laplace type......Page 46
Operators of Dirac type......Page 47
Induced second order boundary conditions......Page 48
The dual operator and the dual boundary condition......Page 49
Green's formula......Page 50
Dirichlet boundary conditions......Page 53
Neumann and Robin boundary conditions......Page 54
Mixed boundary conditions......Page 55
Absolute boundary conditions......Page 56
The de Rham complex and the hodge decomposition theorem......Page 58
Transmission boundary conditions......Page 62
Transmission boundary conditions for the de Rham complex......Page 64
Transfer boundary conditions......Page 65
Bag boundary conditions......Page 66
Spectral boundary conditions......Page 69
Non-minimal operators......Page 73
Oblique boundary conditions......Page 74
Time-dependent heat conduction problems......Page 75
The  rst and second theorems of invariance theory......Page 77
Local invariants of a Riemannian metric......Page 80
The heat trace asymptotics for operators of Laplace type......Page 82
The invariants .........Page 84
Invariance theory......Page 86
Formal cohomology groups of spaces of invariants......Page 89
Expressing the supertrace invariants in terms of a Weyl basis......Page 94
Supertrace invariants for manifolds with boundary......Page 96
Chern-Gauss-Bonnet Theorem......Page 100
Introduction......Page 103
The interior integrands......Page 107
A duality relationship......Page 108
A recursion relation......Page 109
Heat content asymptotics for self-adjoint operators......Page 111
Direct sums......Page 112
Products......Page 113
Dimensional Analysis......Page 116
Relating Dirichlet and Robin boundary conditions......Page 117
Shu e formulae for non-minimal operators......Page 118
Transmission boundary conditions......Page 120
Transfer boundary conditions......Page 122
Time-dependent processes......Page 123
Expressing the invariants .........Page 126
Dimension shifting......Page 129
Spectral boundary conditions......Page 130
Heat content asymptotics for Dirichlet boundary conditions......Page 133
Heat content asymptotics for Robin boundary conditions......Page 140
Proof of Theorem 2.4.1......Page 141
Proof of Theorem 2.4.1......Page 142
Proof of Theorem 2.4.1......Page 143
Heat content asymptotics for mixed boundary conditions......Page 145
The proof of Theorem 2.5.1......Page 146
The proof of Theorem 2.5.2......Page 148
Transmission boundary conditions......Page 150
Proof of Theorem 2.6.1......Page 151
Proof of Theorem 2.6.1......Page 152
Proof of Theorem 2.7.1......Page 154
Proof of Theorem 2.7.1......Page 155
Oblique boundary conditions......Page 156
Variable geometries......Page 159
The proof of Theorem 2.9.1......Page 161
Proof of Theorem 2.9.2......Page 164
Inhomogeneous boundary conditions......Page 166
Heat content asymptotics with interior source p......Page 168
The heat content asymptotics with a boundary heat pump......Page 169
Non-minimal operators......Page 173
Absolute boundary conditions......Page 174
Spectral boundary conditions......Page 175
The proof of Theorem 2.12.1......Page 176
The proof of Theorem 2.12.1......Page 178
Introduction......Page 183
Functorial properties I......Page 185
Heat trace asymptotics for self-adjoint operators......Page 186
Heat trace asymptotics in the 1 dimensional setting......Page 187
Conditions that imply the heat trace coeÆcients are real......Page 188
Product Formulas......Page 189
Dimensional analysis......Page 190
Expressing the invariants en and en;k relative to a Weyl basis......Page 191
Dimension Shifting......Page 192
Variational formulae......Page 194
Recursion relations......Page 195
Transmission boundary conditions......Page 198
Transfer boundary conditions......Page 200
Time dependent processes......Page 202
Spectral boundary conditions......Page 207
Higher order operators......Page 209
Heat trace asymptotics for closed manifolds......Page 210
Heat trace asymptotics for Dirichlet boundary conditions......Page 214
Heat trace asymptotics for Robin boundary conditions......Page 217
Heat trace asymptotics for mixed boundary conditions......Page 222
The proof of Theorem 3.6.1......Page 224
The formula for a5......Page 228
Spectral geometry......Page 230
Proof of Theorem 3.7.1......Page 232
Proof of Theorem 3.7.2......Page 234
Proof of Theorem 3.7.3......Page 235
Proof of Theorem 3.7.4......Page 236
Proof of Theorem 3.7.5......Page 237
Supertrace asymptotics for the Witten Laplacian......Page 238
The proof of Theorem 3.8.2......Page 240
The proof of Theorem 3.8.3......Page 242
Leading terms in the asymptotics......Page 249
Proof of Theorem 3.9.1......Page 251
Proof of Theorem 3.9.2......Page 252
Heat trace asymptotics for transmission boundary conditions......Page 254
Proof of Theorem 3.10.1......Page 256
Heat trace asymptotics for transfer boundary conditions......Page 257
Time-dependent phenomena......Page 261
The eta invariant......Page 269
Spectral boundary conditions......Page 271
Non-minimal operators......Page 275
Fourth order operators......Page 279
Pseudo-di erential operators......Page 283
References......Page 289