Spectral Geometry Of The Laplacian: Spectral Analysis And Differential Geometry Of The Laplacian

by Hajime Urakawa

Overview

The totality of the eigenvalues of the Laplacian of a compact Riemannian manifold is called the spectrum. We describe how the spectrum determines a Riemannian manifold. The continuity of the eigenvalue of the Laplacian, Cheeger and Yau's estimate of the first eigenvalue, the Lichnerowicz-Obata's theorem on the first eigenvalue, the Cheng's estimates of the kth eigenvalues, and Payne-Pólya-Weinberger's inequality of the Dirichlet eigenvalue of the Laplacian are also described. Then, the theorem of Colin de Verdière, that is, the spectrum determines the totality of all the lengths of closed geodesics is described. We give the V Guillemin and D Kazhdan's theorem which determines the Riemannian manifold of negative curvature.

Product Details

ISBN-13:	9789813109087
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	08/02/2017
Pages:	312
Product dimensions:	5.90(w) x 9.10(h) x 0.90(d)

Table of Contents

Preface v

1 Fundamental Materials of Riemaimian Geometry 1

1.1 Introduction 1

1.2 Riemaimian Manifolds 1

1.2.1 Riemannian metrics 1

1.2.2 Lengths of curves 3

1.2.3 Distance 5

1.3 Connection 5

1.3.1 Levi-Civita connection 5

1.3.2 Parallel transport 7

1.3.3 Geodesic 8

1.4 Curvature Tensor Fields 10

1.5 Integration 11

1.6 Divergence of Vector Fields and the Laplacian 12

1.6.1 Divergences of vector fields, gradient vector fields and the Laplacian 12

1.6.2 Green's formula 14

1.7 The Laplacian for Differential Forms 15

1.8 The First and Second Variation Formulas of the Lengths of Curves 17

2 The Space of Riemannian Metrics, and Continuity of the Eigenvalues 21

2.1 Introduction 21

2.2 Symmetric Matrices 21

2.2.1 Eigenvalues of real symmetric matrices 21

2.3 The Space of Riemannian Metrics 28

2.4 Continuity of the Eigenvalues and Upper Semi-continuity of Their Multiplicities 33

2.5 Generic Properties of the Eigenvalues 39

3 Cheeger and Yau Estimates on the Minimum Positive Eigenvalue 53

3.1 Introduction 53

3.2 Main Results of This Chapter 54

3.2.1 Cheeger's estimate for positive minimum eigenvalue λ₂ 54

3.2.2 Yau's estimate of the positive minimum eigenvalue λ₂ 54

3.3 The Co-area Formula 57

3.4 Proofs of Theorems 3.4, 3.5 and Corollary 3.6 62

3.5 Proof of Theorem 3.7 68

3.6 Jacobi Fields and the Comparison Theorem 73

4 The Estimations of the kth Eigenvalue and Lichnerowicz-Obata's Theorem 83

4.1 Introduction 83

4.2 Nodal Domain Theorem Due to R. Courant 83

4.2.1 The boundary problems of the Laplacian 84

4.2.2 Nodal domain theorem of R. Courant 85

4.3 The (Upper Estimates of the kth Eigenvalues 95

4.4 Lichnerowicz-Obata's Theorem 107

5 The Payne, Póly a and Weinberger Type Inequalities for the Dirichlet Eigenvalues 119

5.1 Introduction 119

5.2 Main Results of This Chapter 119

5.3 Preliminary L²-estimates 121

5.4 The Theorem of Cheng and Yang, and Its Corollary 129

5.5 Fundamental Facts on Immersions for Theorem 5.6 133

5.5.1 Isometric immersions and the gradient vector fields 133

5.5.2 Isometric immersion and connections 134

5.5.3 Some lemma on isometric immersion and the Laplacian 135

5.5.4 Proof of Theorem 5.6 139

6 The Heat Equation and the Set of Lengths of Closed Geodesies 143

6.1 Introduction 143

6.2 The Heat Equation on a One-dimensional Circle 144

6.3 Preparation on the Morse Theory 148

6.3.1 Non-degenerate critical submanifolds of Hilbert manifolds 148

6.3.2 Closed geodesies 152

6.3.3 Finite dimensional approximations to Ω(M) 157

6.4 Fundamental Solution of Complex Heat Equation 162

6.5 The Pseudo Fourier Transform 177

6.6 Main Theorems 186

6.7 Several Properties of the Fundamental Solution of the Complex Heat Equation 188

6.8 Mountain Path Method (Stationary Phase Method) 196

6.9 Three Lemmas 207

6.10 Proof of the Main Theorem 6.23 223

7 Negative Curvature Manifolds and the Spectral Rigidity Theorem 229

7.1 Introduction 229

7.2 Spectral Rigidity Theorem Due to Guillemin and Kazhdan 229

7.3 Outline of the Proof of a Spectral Rigidity 231

7.4 The Geodesic Flow Vector Fields 234

7.5 Proof of the Theorem of Livcic 243

7.6 The Space of Harmonic Polynomials, Representation Theory of the Orthogonal Group 252

7.7 The Elliptic Differential Operator on the Space of Symmetric Tensor Fields 263

7.8 Proof of the Main Theorem 7.10 273

7.9 Proofs of the Remaining Three Lemmas 279

7.10 Proof of Spectral Rigidity (Theorem 7.1) 285

Bibliography 291

Index 295