Nonlinear Analysis on Manifolds. Monge-Ampère Equations

ISBN-13: 9780387907048 / Angielski / Twarda / 1982 / 204 str.

Thierry Aubin; T. Aubin

Opis

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 for solving geometrical problems. Conversely, geometry may help us to solve certain problems in analysis. There are several reasons why the topic is difficult and interesting. It is very large and almost unexplored. On the other hand, geometric problems often lead to limiting cases of known problems in analysis, sometimes there is even more than one approach, and the already existing theoretical studies are inadequate to solve them. Each problem has its own particular difficulties. Nevertheless there exist some standard methods which are useful and which we must know to apply them. One should not forget that our problems are motivated by geometry, and that a geometrical argument may simplify the problem under investigation. Examples of this kind are still too rare. This work is neither a systematic study of a mathematical field nor the presentation of a lot of theoretical knowledge. On the contrary, I do my best to limit the text to the essential knowledge. I define as few concepts as possible and give only basic theorems which are useful for our topic. But I hope that the reader will find this sufficient to solve other geometrical problems by analysis.

Spis treści

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. Hölder’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 Kähler Manifolds.- §1. Kähler Manifolds.- 1.1 First Chern Class.- 1.2. Change of Kahler Metrics. Admissible Functions.- §2. Calabi’s Conjecture.- §3. Einstein-Kähler Metrics.- §4. Complex Monge-Ampere Equation.- 4.1. About Regularity.- 4.2. About Uniqueness.- §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.- 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-Ampère 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.