Eigenvalues in Riemannian Geometry (eBook)

Front Cover 1
Eigenvalues in Riemannian Geometry 4
Copyright Page 5
Contents 8
Preface 12
Chapter I. The Laplacian 16
1. Definitions and Preliminaries 16
2. Green’s Formulas 20
3. Basic Facts for Eigenvalue Problems 22
4. The Wave and Heat Equations 26
5. Rayleigh and Max–Min Methods 28
Chapter II. The Basic Examples 41
1. Some Generalities 41
2. Tori 43
3. Weyl’s Formula for Bounded Domains in Rn 46
4. Spheres and Real Projective Spaces 48
5. Disks in Constant Curvature Space Forms 51
Chapter III. .1 and Curvature 70
1. Geodesics and Curvature 70
2. Comparison Theorems for Sectional Curvature Bounded from Above 82
3. Comparison Theorems for Ricci Curvature Bounded from Below 86
4. Obata and Toponogov Theorems 97
Chapter IV. Isoperimetric Inequalities 100
1. The Co-Area Formula 100
2. The Faber–Krahn Inequality 101
3. The Cheeger, Sobolev, and Isoperimetric Constants 110
4. The Sobolev Constant and Eigenvalue, Eigenfunction, Estimates 114
5 .The Constants and Estimates for the Closed Eigenvalue Problem 124
Chapter V. Eigenvalues and the Kinematic Measure 128
1. The Analytic Inequality 128
2. M. Berger’s Isoembolic Inequality 132
3. Cheeger and Isoperimetric Constants, and the Kinematic Measure 139
Chapter VI. The Heat Kernel for Compact Manifolds 149
1. Duhamel’s Principle and Its Consequences 151
2. The Heat Equation on R“ 157
3. The Minakshisundaram-Pleijel Recursion Formulas 163
4. Existence of the Heat Kernel 166
Chapter VII. The Dirichlet Heat Kernel for Regular Domains 173
1. Preliminaries 174
2. The Dirichlet Heat Kernel for Regular Domains 176
3. Duhamel’s Principle 179
Chapter VIII. The Heat Kernel for Noncompact Manifolds 194
1. The Maximum Principle, and Uniqueness Theorems, for the Heat Operator 195
2. The Heat Kernel for Noncompact Manifolds 202
3. Comparison Theorems for Heat Kernels 207
4. Upper Bounds for the Heat Kernel 166
Chapter IX. Topological Perturbations with Negligible Effect 222
1. Statement and Discussion of the Results 224
2. Proof of Theorems 1–6 231
3. Using Rayleigh’s Characterization of Eigenvalues 243
4. The Mathematics of Crushed Ice 248
Chapter X. Surfaces of Constant Negative Curvature 254
1. Geometry of the Hyperbolic Plane 254
2. The Heat Kernel of the Hyperbolic Plane 257
3. Löbell Surfaces and the Estimates of P. Buser 261
4. Low Eigenvalues and Short Geodesics 269
5. The Upper Half-Space Model of Hyperbolic Space 277
Chapter XI. The Selberg Trace Formula 281
1. Preliminaries 281
2. The Pretrace Formula 286
3. Applications of the Pretrace Formula 293
4. Low Eigenvalues and Short Geodesics 269
5. Applications of the Trace Formula 308
6. Concluding Remarks 317
Chapter XII. Miscellanea 318
1. Volumes of Disks and Spheres 318
2. The Fourier Transform 319
3. The Poisson Summation Formula 321
4. The Fourier Transform and the Heat Equation 322
5. Eigenfunctions on Spheres and Hyperbolic Space 323
6. Minimal Submanifolds of Euclidean Spaces and Spheres 324
7. Normalization of Geometric Data 329
8. Geodesic Coordinates 331
9. The Levy–Gromov Isopenmetric Inequality 337
10. Heat Conduction on the Euclidean Upper Half-Space 340
11. The Maximum Principle for the Laplacian 344
12. Recent Progress on Eigenvalue and Heat Kernel Estimates 345
Appendix: Laplacian on Forms 349
Bibliography 360
Index 374
Pure and Applied Mathematics 378