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Lectures on differential geometry / Richard Schoen and Shing-Tung Yau.

By: Schoen, Richard (Richard M.).
Contributor(s): Yau, Shing-Tung, 1949-.
Material type: materialTypeLabelBookSeries: Conference proceedings and lecture notes in geometry and topology ; v. 1. Publisher: Cambridge, MA : International Press, c1994, 2010Description: v, 414 p. ; 24 cm.ISBN: 1571460128 (acidfree paper); 9781571461988 (paperback reissue).Subject(s): Geometry, DifferentialOther classification: 53-01 (53-02 53C21 58G30)
Contents:
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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