Nonlinear Analysis on Manifolds. Monge-Ampère Equations

1 Riemannian Geometry.- §1. Introduction to Differential Geometry.- §2. Riemannian Manifold.- §3. Exponential Mapping.- §4. The Hopf-Rinow Theorem.- §5. Second Variation of the Length Integral.- §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.- §11. Harmonic Forms.- 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.- §2. Four Basic Theorems of Functional Analysis.- §3. Weak Convergence. Compact Operators.- §4. The Lebesgue Integral.- §5. The LpSpaces.- §6. Elliptic Differential Operators.- §7. Inequalities.- §8. Maximum Principle.- §9. Best Constants.- 4 Green’s Function for Riemannian Manifolds.- §1. Linear Elliptic Equations.- §2. Green’s Function of the Laplacian.- 5 The Methods.- §1. Yamabe’s Equation.- §2. Berger’s Problem.- §3. Nirenberg’s Problem.- 6 The Scalar Curvature.- §1. The Yamabe Problem.- §2. The Positive Case.- §3. Other Problems.- 7 Complex Monge-Ampere Equation on Compact Kähler Manifolds.- §1. Kähler Manifolds.- §2. Calabi’s Conjecture.- §3. Einstein-Kähler Metrics.- §4. Complex Monge-Ampere Equation.- §5. Theorem of Existence (the Negative Case).- §6. Existence of Kähler-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-Ampère Equations.- §1. Monge-Ampère Equations on Bounded Domains of ?n.- §2. The Estimates.- §3. The Radon Measure ?(?).- §4. The Functional ? (?).- §5. Variational Problem.- §6. The Complex Monge-Ampère Equation.- §7. The Case of Radially Symmetric Functions.- §8. A New Method.- Notation.