Lectures on differential geometry / Richard Schoen and Shing-Tung Yau.
Idioma: Inglés Lenguaje original: Chino Series Conference proceedings and lecture notes in geometry and topology ; v. 1Editor: Cambridge, MA : International Press, c1994, 2010Descripción: v, 414 p. ; 24 cmISBN: 1571460128 (acidfree paper); 9781571461988 (paperback reissue)Tema(s): Geometry, DifferentialOtra clasificación: 53-01 (53-02 53C21 58G30)Table of Contents Preface for the English Translation i Translation of Original Preface ii Chapter I: Comparison Theorems and Gradient Estimates [1] §1. Comparison Theorems [1] §2. Splitting Theorem [12] §3. Gradient Estimate [17] §4. Complete Riemannian Manifolds of Non-Negative Ricci Curvature [23] Chapter II: Harmonic Functions on Manifolds with Negative Curvature [31] §1. Geometric Boundary S(∞) and Solvability of the Dirichlet Problem [32] §2. Harnack Inequality and Poisson Kernel [41] §3. Martin Boundary and Martin Integral Representation [50] §4. Proof of Harnack Inequalities [55] §5. Harmonic Functions on More General Manifolds [65] §6. Mean Value Inequality for Subharmonic Functions [75] Appendix to Chapter II: The Existence of an Entire Green’s Function [81] Chapter III: Eigenvalue Problems [87] §1. Basic Properties of Eigenvalues [87] §2. The Heat Kernel of Riemannian Manifolds [93] §3. Upper Bounds for the First Eigenvalue λ1 [104] §4. Lower Bounds for the First Eigenvalue λ1 [106] §5. Estimates on Higher Eigenvalues [117] §6. Nodal Sets and Multiplicities of Eigenvalues [122] §7. Gaps Between Eigenvalues [128] §8. Eigenvalue Problems for Surfaces [134] Chapter IV: Heat Kernel on Riemannian Manifolds [155] §1. Gradient Estimates of Heat Kernel [155] §2. Harnack Inequality and Estimates for the Heat Kernel [163] §3. Applications of the Estimates for Heat Kernel [176] Chapter V: Conformal Deformation of Scalar Curvatures [183] §1. The Two-Dimensional Case [187] §2. Yamabe Problem and Conformal Invariant λ(M) [199] §3. Conformal Normal Coordinates and Asymptotic Expansion of Green’s Function [207] §4. The Resolution of Yamabe Problem [219] Appendix to Chapter V: Best Constant in the Sobolev Inequality [224] Chapter VI: Locally Conformally Flat Manifolds [231] §1. Conformal Transformations and Locally Conformally Flat Manifolds [232] §2. Conformal Invariants [239] §3. Embeddings of Locally Conformally Flat Manifolds into Sn [253] §4. Topology of Locally Conformally Flat Manifolds [263] §5. P.D.E. Aspects of the Theory [273] Chapter VII: Problem Section [277] §1. Curvature and the Topology of Manifolds [278] §11. Curvature and the Complex Structure [285] §111. Submanifolds [288] §IV. The Spectrum [293] §V. Problems Related to Geodesics [296] §VI. Minimal Submanifolds [297] §VII. General Relativity and the Yang-Mills Equation [303] Bibliography [305] Chapter VIII: Nonlinear Analysis in Geometry [315] §1. Eigenvalues and Harmonic Functions [318] §2. Yamabe’s Equation and Conformally Flat Manifolds [324] §3. Harmonic Maps [326] §4. Minimal Submanifolds [330] §5. Kahler Geometry [335] §6. Canonical Metrics over Complex Manifolds [344] References [358] Chapter IX: Open Problems in Differential Geometry [365] §1. Metric Geometry [365] §11. Classical Euclidean Geometry [372] J1ll. Partial Differential Equations [380] §IV. Kahler Geometry [387] References [405] Bibliography [411] Index [415]
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Incluye referencias bibliográficas e índice.
Table of Contents --
Preface for the English Translation i --
Translation of Original Preface ii --
Chapter I: Comparison Theorems and Gradient Estimates [1] --
§1. Comparison Theorems [1] --
§2. Splitting Theorem [12] --
§3. Gradient Estimate [17] --
§4. Complete Riemannian Manifolds of Non-Negative Ricci Curvature [23] --
Chapter II: Harmonic Functions on Manifolds with Negative Curvature [31] --
§1. Geometric Boundary S(∞) and Solvability of the Dirichlet Problem [32] --
§2. Harnack Inequality and Poisson Kernel [41] --
§3. Martin Boundary and Martin Integral Representation [50] --
§4. Proof of Harnack Inequalities [55] --
§5. Harmonic Functions on More General Manifolds [65] --
§6. Mean Value Inequality for Subharmonic Functions [75] --
Appendix to Chapter II: The Existence of an Entire Green’s Function [81] --
Chapter III: Eigenvalue Problems [87] --
§1. Basic Properties of Eigenvalues [87] --
§2. The Heat Kernel of Riemannian Manifolds [93] --
§3. Upper Bounds for the First Eigenvalue λ1 [104] --
§4. Lower Bounds for the First Eigenvalue λ1 [106] --
§5. Estimates on Higher Eigenvalues [117] --
§6. Nodal Sets and Multiplicities of Eigenvalues [122] --
§7. Gaps Between Eigenvalues [128] --
§8. Eigenvalue Problems for Surfaces [134] --
Chapter IV: Heat Kernel on Riemannian Manifolds [155] --
§1. Gradient Estimates of Heat Kernel [155] --
§2. Harnack Inequality and Estimates for the Heat Kernel [163] --
§3. Applications of the Estimates for Heat Kernel [176] --
Chapter V: Conformal Deformation of Scalar Curvatures [183] --
§1. The Two-Dimensional Case [187] --
§2. Yamabe Problem and Conformal Invariant λ(M) [199] --
§3. Conformal Normal Coordinates and Asymptotic Expansion of Green’s Function [207] --
§4. The Resolution of Yamabe Problem [219] --
Appendix to Chapter V: Best Constant in the Sobolev Inequality [224] --
Chapter VI: Locally Conformally Flat Manifolds [231] --
§1. Conformal Transformations and Locally Conformally Flat Manifolds [232] --
§2. Conformal Invariants [239] --
§3. Embeddings of Locally Conformally Flat Manifolds into Sn [253] --
§4. Topology of Locally Conformally Flat Manifolds [263] --
§5. P.D.E. Aspects of the Theory [273] --
Chapter VII: Problem Section [277] --
§1. Curvature and the Topology of Manifolds [278] --
§11. Curvature and the Complex Structure [285] --
§111. Submanifolds [288] --
§IV. The Spectrum [293] --
§V. Problems Related to Geodesics [296] --
§VI. Minimal Submanifolds [297] --
§VII. General Relativity and the Yang-Mills Equation [303] --
Bibliography [305] --
Chapter VIII: Nonlinear Analysis in Geometry [315] --
§1. Eigenvalues and Harmonic Functions [318] --
§2. Yamabe’s Equation and Conformally Flat Manifolds [324] --
§3. Harmonic Maps [326] --
§4. Minimal Submanifolds [330] --
§5. Kahler Geometry [335] --
§6. Canonical Metrics over Complex Manifolds [344] --
References [358] --
Chapter IX: Open Problems in Differential Geometry [365] --
§1. Metric Geometry [365] --
§11. Classical Euclidean Geometry [372] --
J1ll. Partial Differential Equations [380] --
§IV. Kahler Geometry [387] --
References [405] --
Bibliography [411] --
Index [415] --
MR, MR1333601
Translated from the Chinese.
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