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E-raamat: Spectral Geometry Of The Laplacian: Spectral Analysis And Differential Geometry Of The Laplacian

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(Tohoku Univ, Japan)
  • Formaat: 312 pages
  • Ilmumisaeg: 02-Jun-2017
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789813109100
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Spectral Geometry Of The Laplacian: Spectral Analysis And Differential Geometry Of The LaplacianSuurem pilt
  • Formaat: 312 pages
  • Ilmumisaeg: 02-Jun-2017
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789813109100
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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.
Preface v
1 Fundamental Materials of Riemannian Geometry
1(20)
1.1 Introduction
1(1)
1.2 Riemannian Manifolds
1(4)
1.2.1 Riemannian metrics
1(2)
1.2.2 Lengths of curves
3(2)
1.2.3 Distance
5(1)
1.3 Connection
5(5)
1.3.1 Levi-Civita connection
5(2)
1.3.2 Parallel transport
7(1)
1.3.3 Geodesic
8(2)
1.4 Curvature Tensor Fields
10(1)
1.5 Integration
11(1)
1.6 Divergence of Vector Fields and the Laplacian
12(3)
1.6.1 Divergences of vector fields, gradient vector fields and the Laplacian
12(2)
1.6.2 Green's formula
14(1)
1.7 The Laplacian for Differential Forms
15(2)
1.8 The First and Second Variation Formulas of the Lengths of Curves
17(4)
2 The Space of Riemannian Metrics, and Continuity of the Eigenvalues
21(32)
2.1 Introduction
21(1)
2.2 Symmetric Matrices
21(7)
2.2.1 Eigenvalues of real symmetric matrices
21(7)
2.3 The Space of Riemannian Metrics
28(5)
2.4 Continuity of the Eigenvalues and Upper Semi-continuity of Their Multiplicities
33(6)
2.5 Generic Properties of the Eigenvalues
39(14)
3 Cheeger and Yau Estimates on the Minimum Positive Eigenvalue
53(30)
3.1 Introduction
53(1)
3.2 Main Results of This
Chapter
54(3)
3.2.1 Cheeger's estimate for positive minimum eigenvalue λ2
54(1)
3.2.2 Yau's estimate of the positive minimum eigenvalue λ2
54(3)
3.3 The Co-area Formula
57(5)
3.4 Proofs of Theorems 3.4, 3.5 and Corollary 3.6
62(6)
3.5 Proof of Theorem 3.7
68(5)
3.6 Jacobi Fields and the Comparison Theorem
73(10)
4 The Estimations of the kth Eigenvalue and Lichnerowicz-Obata's Theorem
83(36)
4.1 Introduction
83(1)
4.2 Nodal Domain Theorem Due to R. Courant
83(12)
4.2.1 The boundary problems of the Laplacian
84(1)
4.2.2 Nodal domain theorem of R. Courant
85(10)
4.3 The Upper Estimates of the kth Eigenvalues
95(12)
4.4 Lichnerowicz-Obata's Theorem
107(12)
5 The Payne, Polya and Weinberger Type Inequalities for the Dirichlet Eigenvalues
119(24)
5.1 Introduction
119(1)
5.2 Main Results of This
Chapter
119(2)
5.3 Preliminary L2-estimates
121(8)
5.4 The Theorem of Cheng and Yang, and Its Corollary
129(4)
5.5 Fundamental Facts on Immersions for Theorem 5.6
133(10)
5.5.1 Isometric immersions and the gradient vector fields
133(1)
5.5.2 Isometric immersion and connections
134(1)
5.5.3 Some lemma on isometric immersion and the Laplacian
135(4)
5.5.4 Proof of Theorem 5.6
139(4)
6 The Heat Equation and the Set of Lengths of Closed Geodesics
143(86)
6.1 Introduction
143(1)
6.2 The Heat Equation on a One-dimensional Circle
144(4)
6.3 Preparation on the Morse Theory
148(14)
6.3.1 Non-degenerate critical submanifolds of Hilbert manifolds
148(4)
6.3.2 Closed geodesics
152(5)
6.3.3 Finite dimensional approximations to Ω(M)
157(5)
6.4 Fundamental Solution of Complex Heat Equation
162(15)
6.5 The Pseudo Fourier Transform
177(9)
6.6 Main Theorems
186(2)
6.7 Several Properties of the Fundamental Solution of the Complex Heat Equation
188(8)
6.8 Mountain Path Method (Stationary Phase Method)
196(11)
6.9 Three Lemmas
207(16)
6.10 Proof of the Main Theorem 6.23
223(6)
7 Negative Curvature Manifolds and the Spectral Rigidity Theorem
229(62)
7.1 Introduction
229(1)
7.2 Spectral Rigidity Theorem Due to Guillemin and Kazhdan
229(2)
7.3 Outline of the Proof of a Spectral Rigidity
231(3)
7.4 The Geodesic Flow Vector Fields
234(9)
7.5 Proof of the Theorem of Livcic
243(9)
7.6 The Space of Harmonic Polynomials, Representation Theory of the Orthogonal Group
252(11)
7.7 The Elliptic Differential Operator on the Space of Symmetric Tensor Fields
263(10)
7.8 Proof of the Main Theorem 7.10
273(6)
7.9 Proofs of the Remaining Three Lemmas
279(6)
7.10 Proof of Spectral Rigidity (Theorem 7.1)
285(6)
Bibliography 291(4)
Index 295
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