Monge-Ampere equations and microlocal analysis on Kahler manifolds

AbstractAward：DMS 1206284，首席研究员：Yanir Rubinstein该项目主要关注微分几何和几何分析中的问题，这些问题可以用公式表示为真实的和复杂的Monge-Ampere型方程。其中包括：（1）具有锥奇点的Kahler-Einstein度量的存在性和正则性及其应用;（2）齐次Monge-Ampere方程解的存在性和正则性及其适定性;（3）凸几何中出现的新的Monge-Ampere型方程及其与偏微分方程和Legendre变换的关系。该项目的一个关键部分的特点是结合联合收割机的微局部分析工具来研究这些方程，除了更传统的偏微分方程，凸分析，多能理论，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，在这个建议中开发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，加深我们对卡勒流形上正则几何的理解，似乎是试图模拟宇宙几何的物理学家感兴趣的。另一方面，这些规范几何与数学中各种各样的既定领域有关系。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，并且具有广泛的现实应用，例如气象学和网络的优化设计。开发方法和技术来构造和近似解这些方程，并研究其规律性，可以在这些方程出现的其他情况下应用。最后，勒让德变换是数学、力学和经济学中的经典工具，寻求将这一理论推广到其他环境，如本项目，可以找到广泛的应用。

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.