Nonlinear Analysis on Manifolds. Monge-Ampère Equations

1. Front Matter

   Pages i-xii

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2. Riemannian Geometry

   • Thierry Aubin

   Pages 1-31

3. Sobolev Spaces

   • Thierry Aubin

   Pages 32-69

4. Background Material

   • Thierry Aubin

   Pages 70-100

5. Green’s Function for Riemannian Manifolds

   • Thierry Aubin

   Pages 101-114

6. The Methods

   • Thierry Aubin

   Pages 115-124

7. The Scalar Curvature

   • Thierry Aubin

   Pages 125-138

8. Complex Monge-Ampère Equation on Compact Kähler Manifolds

   • Thierry Aubin

   Pages 139-156

9. Monge-Ampère Equations

   • Thierry Aubin

   Pages 157-188

10. Back Matter

    Pages 189-204

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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.

• Eigenvalue
• Interpolation
• Jacobi field
• Riemannian geometry
• Riemannian manifold
• Tensor
• curvature
• differential geometry
• manifold
• matrix theory

• Universite de Paris VI Mathematiques, Paris, Cedex 05, France

  Thierry Aubin