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. Language: English