Nonlinear Analysis on Manifolds. Monge-Ampère Equations

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Monge-Ampère equation - Encyclopedia of Mathematics

Article Navigation. Close mobile search navigation Article Navigation. Article Contents. Oxford Academic. Google Scholar. Jiayu Li. Xi Zhang. Correspondence to be sent to: e-mail: mathzx ustc. Cite Citation. Permissions Icon Permissions.

Abstract In this paper, we prove a mean value formula for bounded subharmonic Hermitian matrix valued function on a complete Riemannian manifold with nonnegative Ricci curvature. In this paper, we study a class of Hermitian matrix valued functions and establish a mean value theorem for them. Applying Theorem 1.

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Search ADS. Intrinsic norms. The Dirichlet problem for nonlinear second-order elliptic equations. Differential equations on Riemannian manifolds and their geometric applications.

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Linear growth harmonic functions on complete manifolds with nonnegative Ricci curvature. Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature. Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals. Issue Section:. Download all figures. View Metrics. Email alerts New issue alert. Advance article alerts. Article activity alert. Actions Shares.

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Nonlinear Analysis on Manifolds. Aubin 2. Publisher : Springer Release Date : 3. 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.

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Nonlinear Analysis on Manifolds. Monge-Ampere Equations

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.

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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. You just clipped your first slide! Clipping is a handy way to collect important slides you want to go back to later. Now customize the name of a clipboard to store your clips.

Complex Monge-Ampere Equations (Day 1) - Valentino Tosatti