Monge-Ampere equations and microlocal analysis on Kahler manifolds

Monge-Ampere 方程和 Kahler 流形上的微局域分析

• 批准号: 1206284
• 负责人: Yanir Rubinstein
• 金额: $ 12万
• 依托单位: American Institute of Mathematics
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1206284&HistoricalAwards=false
• 关键词: Monge; Ampere; equations; microlocal; analysis

AbstractAward: DMS 1206284, Principal Investigator: Yanir RubinsteinThis project focuses on problems mainly in differential geometry and geometric analysis that can be formulated as real and complex Monge-Ampere type equations. These include (1) the existence and regularity of Kahler-Einstein metrics with conic singularities and their applications; (2) the existence and regularity of solutions to and well-posedness of the homogeneous Monge-Ampere equation; (3) new equations of Monge-Ampere type that arise in convex geometry and their relations to PDEs and the Legendre transform. One feature in key parts of this project is to combine tools of microlocal analysis to study these equations, in addition to the more traditional methods of PDEs, convex analysis, pluripotential theory, and several complex variables. Another theme is to investigate novel relations between convex analysis and geometry and complex analysis and geometry.In general terms, the analytic techniques developed in this proposal should be useful to researchers working in geometry, physics and elsewhere. On the one hand, deepening our understanding of canonical geometries on Kahler manifolds seems to be of interest to physicists trying to model the geometry of the universe. On the other hand, these canonical geometries have relations to a wide variety of established fields in mathematics. Moreover, Monge-Ampere type equations arise in a wide variety of problems in pure and applied mathematics and have a wide range of real-world applications, such as meteorology and optimal design of networks. Developing methods and techniques to construct and approximate solutions to such equations and to study their regularity could have applications in other instances where these equations appear. Finally, the Legendre transform is a classical tool in mathematics, mechanics and economics, and seeking generalizations of this theory to other settings, as in this project, could find a broad range of applications.

该项目主要关注微分几何和几何分析中的问题，这些问题可以表示为实数和复数的Monge-Ampere型方程。其中包括(1)具有二次奇性的Kahler-Einstein度量的存在性、正则性及其应用；(2)齐次Monge-Ampere方程解的存在性、正则性和适定性；(3)凸几何中出现的新的Monge-Ampere型方程及其与偏微分方程组和Legendre变换的关系。这个项目的关键部分的一个特点是结合微局部分析的工具来研究这些方程，除了更传统的偏微分方程组、凸分析、多势理论和几个复变量的方法。另一个主题是研究凸分析与几何和复分析与几何之间的新关系。总的来说，这项建议中发展的分析技术应该对几何、物理和其他领域的研究人员有用。一方面，加深我们对Kahler流形上正则几何的理解似乎是试图对宇宙几何建模的物理学家感兴趣的。另一方面，这些正则几何与数学中的许多既定领域都有关系。此外，Monge-Ampere型方程出现在纯数学和应用数学中的各种问题中，并在现实世界中有广泛的应用，如气象学和网络优化设计。发展方法和技术来构造和逼近这些方程的解，并研究它们的正则性，可以在出现这些方程的其他情况下应用。最后，勒让德变换是数学、力学和经济学中的经典工具，寻求将该理论推广到其他环境，如本项目，可以找到广泛的应用。