Nonlinear analysis on manifolds, Monge-Ampère equations

1 Riemannian Geometry.- 1. Introduction to Differential Geometry.- 1.1 Tangent Space.- 1.2 Connection.- 1.3 Curvature.- 2. Riemannian Manifold.- 2.1 Metric Space.- 2.2 Riemannian Connection.- 2.3 Sectional Curvature. Ricci Tensor. Scalar Curvature.- 2.4 Parallel Displacement. Geodesic.- 3. Exponential Mapping.- 4. The Hopf-Rinow Theorem.- 5. Second Variation of the Length Integral.- 5.1 Existence of Tubular Neighborhoods.- 5.2 Second Variation of the Length Integral.- 5.3 Myers' Theorem.- 6. Jacobi Field.- 7. The Index Inequality.- 8. Estimates on the Components of the Metric Tensor.- 9. Integration over Riemannian Manifolds.- 10. Manifold with Boundary.- 10.1. Stokes' Formula.- 11. Harmonic Forms.- 11.1. Oriented Volume Element.- 11.2. Laplacian.- 11.3. Hodge Decomposition Theorem.- 11.4. Spectrum.- 2 Sobolev Spaces.- 1. First Definitions.- 2. Density Problems.- 3. Sobolev Imbedding Theorem.- 4. Sobolev's Proof.- 5. Proof by Gagliardo and Nirenberg.- 6. New Proof.- 7. Sobolev Imbedding Theorem for Riemannian Manifolds.- 8. Optimal Inequalities.- 9. Sobolev's Theorem for Compact Riemannian Manifolds with Boundary.- 10. The Kondrakov Theorem.- 11. Kondrakov's Theorem for Riemannian Manifolds.- 12. Examples.- 13. Improvement of the Best Constants.- 14. The Case of the Sphere.- 15. The Exceptional Case of the Sobolev Imbedding Theorem.- 16. Moser's Results.- 17. The Case of the Riemannian Manifolds.- 18. Problems of Traces.- 3 Background Material.- 1. Differential Calculus.- 1.1. The Mean Value Theorem.- 1.2. Inverse Function Theorem.- 1.3. Cauchy's Theorem.- 2. Four Basic Theorems of Functional Analysis.- 2.1. Hahn-Banach Theorem.- 2.2. Open Mapping Theorem.- 2.3. The Banach-Steinhaus Theorem.- 2.4. Ascoli's Theorem.- 3. Weak Convergence. Compact Operators.- 3.1. Banach's Theorem.- 3.2. The Leray-Schauder Theorem.- 3.3. The Fredholm Theorem.- 4. The Lebesgue Integral.- 4.1. Dominated Convergence Theorem.- 4.2. Fatou's Theorem.- 4.3. The Second Lebesgue Theorem.- 4.4. Rademacher's Theorem.- 4.5. Fubini's Theorem.- 5. The LpSpaces.- 5.1. Regularization.- 5.2. Radon's Theorem.- 6. Elliptic Differential Operators.- 6.1. Weak Solution.- 6.2. Regularity Theorems.- 6.3. The Schauder Interior Estimates.- 7. Inequalities.- 7.1. Holder's Inequality.- 7.2. Clarkson's Inequalities.- 7.3. Convolution Product.- 7.4. The Calderon-Zygmund Inequality.- 7.5. Korn-Lichtenstein Theorem.- 7.6. Interpolation Inequalities.- 8. Maximum Principle.- 8.1. Hopf's Maximum Principle.- 8.2. Uniqueness Theorem.- 8.3. Maximum Principle for Nonlinear Elliptic Operator of Order Two.- 8.4. Generalized Maximum Principle.- 9. Best Constants.- 9.1. Application to Sobolev Spaces.- 4 Green's Function for Riemannian Manifolds.- 1. Linear Elliptic Equations.- 1.1. First Nonzero Eigenvalue ? of ?.- 1.2. Existence Theorem for the Equation ?? = f.- 2. Green's Function of the Laplacian.- 2.1. Parametrix.- 2.2. Green's Formula.- 2.3. Green's Function for Compact Manifolds.- 2.4. Green's Function for Compact Manifolds with Boundary.- 5 The Methods.- 1. Yamabe's Equation.- 1.1. Yamabe's Method.- 2. Berger's Problem.- 2.1. The Positive Case.- 3. Nirenberg's Problem.- 3.1. A Nonlinear Theorem of Fredholm.- 3.2. Open Questions.- 6 The Scalar Curvature.- 1. The Yamabe Problem.- 1.1. Yamabe's Functional.- 1.2. Yamabe's Theorem.- 2. The Positive Case.- 2.1. Geometrical Applications.- 2.2. Open Questions.- 3. Other Problems.- 3.1. Topological Meaning of the Scalar Curvature.- 3.2. Kazdan and Warner's Problem.- 7 Complex Monge-Ampere Equation on Compact Kahler Manifolds.- 1. Kahler Manifolds.- 1.1 First Chern Class.- 1.2. Change of Kahler Metrics. Admissible Functions.- 2. Calabi's Conjecture.- 3. Einstein-Kahler Metrics.- 4. Complex Monge-Ampere Equation.- 4.1. About Regularity.- 4.2. About Uniqueness.- 5. Theorem of Existence (the Negative Case).- 6. Existence of Kahler-Einstein Metric.- 7. Theorem of Existence (the Null Case).- 8. Proof of Calabi's Conjecture.- 9. The Positive Case.- 10. A Priori Estimate for ??.- 11. A Priori Estimate for the Third Derivatives of Mixed Type.- 12. The Method of Lower and Upper Solutions.- 8 Monge-Ampere Equations.- 1. Monge-Ampere Equations on Bounded Domains of ?n.- 1.1. The Fundamental Hypothesis.- 1.2. Extra Hypothesis.- 1.3. Theorem of Existence.- 2. The Estimates.- 2.1. The First Estimates.- 2.2. C2-Estimate.- 2.3. C3-Estimate.- 3. The Radon Measure ?(?).- 4. The Functional ? (?).- 4.1. Properties of ? (?).- 5. Variational Problem.- 6. The Complex Monge-Ampere Equation.- 6.1. Bedford's and Taylor's Results.- 6.2. The Measure M(?).- 6.3. The Functional J(?).- 6.4. Some Properties of J(?).- 7. The Case of Radially Symmetric Functions.- 7.1. Variational Problem.- 7.2. An Open Problem.- 8. A New Method.- Notation.