Microlocal Analysis and Monge-Ampere Type Equations in Geometry

几何中的微局域分析和Monge-Ampere型方程

• 批准号: 1515703
• 负责人: Yanir Rubinstein
• 金额: $ 19.45万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1515703&HistoricalAwards=false
• 关键词: Microlocal; Analysis; Monge; Ampere; Type

This project focuses on problems mainly in geometric analysis that can be formulated as partial differential equations of Monge-Ampere type, broadly understood. In general terms, the analytic and geometric 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 and Lagrangians in Calabi-Yau 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.A number of the equations proposed in this project are new, fall outside of the traditionally known Monge-Ampere type equations, and have exciting new geometric applications requiring new analytical tools. A novel feature in this research is to apply several different tools of microlocal analysis, traditionally pertaining to linear problems, to study these fully nonlinear equations. Another theme is to investigate novel relations between convex analysis and geometry and complex analysis and geometry. The problems investigated include: (1) Kahler-Einstein metrics with conic singularities. These metrics provide a new powerful analytic tool in algebraic and complex geometry. (2) The PI will study a new degenerate version of the special Lagrangian equation. It governs geodesics in the space of positive Lagrangians on a Calabi-Yau manifold. Understanding solutions of this equation requires new methods and will have applications to existence and uniqueness of special Lagrangians, singularities of Lagrangian mean curvature flow, the topology of the space of Lagrangians, and Lagrangian intersection theory. (3) The PI shall study the space of Kahler metrics using geodesics in the space, finite-dimensional Bergman approximations, Fourier integral operators with complex phase, and the metric space geometry of this space. (4) The PI is developing interactions between complex and convex geometry, including a differential theory for the polarity transform in parallel to the known theory for the Legendre transform. New equations of Hamilton-Jacobi and Monge-Ampere type that arise from this will be investigated as well as complex analogues. These equations provide new processes for interpolation between Banach spaces and new notions of optimal transportation.

本项目主要研究几何分析中的问题，这些问题可以被广泛理解为蒙日-安培型偏微分方程。一般来说，在这个提议中发展的解析和几何技术应该对研究几何、物理和其他领域的研究人员有用。一方面，加深我们对Kahler流形的正则几何和Calabi-Yau流形中的拉格朗日的理解似乎对试图模拟宇宙几何的物理学家很有兴趣。另一方面，这些正则几何与数学中许多已建立的领域都有关系。此外，蒙日-安培型方程出现在纯数学和应用数学的各种问题中，并具有广泛的现实应用，例如气象学和网络优化设计。开发方法和技术来构造和近似求解这些方程，并研究它们的规律性，可以应用于这些方程出现的其他实例。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，就像在这个项目中一样，可以找到广泛的应用。在这个项目中提出的许多方程是新的，不属于传统上已知的蒙日-安培型方程，并且具有令人兴奋的新的几何应用，需要新的分析工具。本研究的一个新特点是应用几种不同的微局部分析工具，传统上用于线性问题，来研究这些完全非线性方程。另一个主题是研究凸分析与几何、复分析与几何之间的新关系。研究的问题包括：(1)具有二次奇点的Kahler-Einstein度量。这些度量为代数和复杂几何提供了一种新的强大的分析工具。(2) PI将研究特殊拉格朗日方程的新简并版本。它支配Calabi-Yau流形上正拉格朗日空间中的测地线。理解这个方程的解需要新的方法，并将应用于特殊拉格朗日量的存在唯一性、拉格朗日平均曲率流的奇异性、拉格朗日空间的拓扑结构以及拉格朗日交点理论。(3) PI应使用空间中的测大地线、有限维Bergman近似、复相位傅里叶积分算子以及该空间的度量空间几何来研究Kahler度量空间。(4) PI正在发展复杂几何和凸几何之间的相互作用，包括与已知的勒让德变换理论并行的极性变换微分理论。由此产生的Hamilton-Jacobi和Monge-Ampere型新方程以及复杂的类似方程将被研究。这些方程提供了巴拿赫空间间插值的新过程和最优运输的新概念。