Nonlinear Analysis on Manifolds. Monge-Ampère Equations
This volume is intended to allow mathematicians and physicists, especially analysts, to learn about nonlinear problems which arise in Riemannian Geometry. Analysis on Riemannian manifolds is a field currently undergoing great development. More and more, analysis proves to be a very powerful means fo...
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| Format: | eBook |
| Language: | English |
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New York, NY
Springer New York
1982, 1982
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| Edition: | 1st ed. 1982 |
| Series: | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
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| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 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
- §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
- §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