Differential Geometry and Partial Differential Equations

微分几何和偏微分方程

• 批准号: 1540379
• 负责人: Richard Schoen
• 金额: $ 24.29万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-10-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1540379&HistoricalAwards=false
• 关键词: Differential; Geometry; Partial; Equations; Differential; Geometry; Partial; Equations

One aspect of the proposed research has to do with optimal shapes of surfaces. If we think of the surface as a drumhead which vibrates freely at certain frequencies, then, roughly speaking, the more complicated a geometry we have the smaller its fundamental frequencies will be. This suggests the problem of looking for geometries which maximize the fundamental frequency for their area. This extremal question is a difficult and much studied problem. It turns out that the geometries which arise are related to surfaces of least area (soap films). The proposer will investigate such extremal configurations for surfaces with boundary. The other main area of investigation concerns the Einstein equations of general relativity. These equations describe the gravitational field for massive bodies in the universe. The theory is purely geometric and is a wave theory with an initial value formulation. The proposer is planning to investigate the geometry of solutions to give conditions under which gravitational collapse takes place and black holes are formed. Such questions lead to important geometric questions involving gravitational energy and curvature of spacetime.The proposed research is at the interface between differential geometry, general relativity, and partial differential equations. A main theme of the research in geometry will be the study of spectral geometry. The PI plans to construct metrics on surfaces and certain higher dimensional manifolds subject to an area or boundary length constraint which maximize the first eigenvalue. This is a nonstandard type of variational problem since it involves maximizing and minimizing over infinite dimensional spaces of competitors. The PI will study the geometry of such maximizing metrics to determine the optimal shapes with largest fundamental frequency. In relativity, the PI intends to continue his investigations into the construction proposed by Bartnik of mass minimizing extensions of compact domains and static vacuum metrics. The PI also intends to study geometric properties of initial data sets concerning the question of whether they can contain non-compact stable trapped surfaces. Finally the PI intends to investigate global properties of the moduli space of solutions of the constraint equations which define the possible initial data for the Einstein equations. The PI shall pursue a range of questions concerning minimal submanifolds satisfying free boundary conditions and connections to eigenvalue problems. Finally the PI plans to continue his study of minimal lagrangian and special lagrangian submanifolds of Kahler-Einstein manifolds. The research will attempt to prove a conjecture concerning the invariance of the subgroup of the integral homology of a Calabi-Yau manifold which is generated by minimal lagrangian cycles when one deforms the ambient Calabi-Yau structure.

拟议的研究的一个方面与最佳表面形状有关。如果我们将表面视为在某些频率下自由振动的鼓头，那么大致来说，几何形状越复杂，我们的基本频率就越小。这表明寻找几何形状，从而最大程度地提高其区域的基本频率。这个极端问题是一个困难而又研究的问题。事实证明，出现的几何形状与面积最小的表面（肥皂膜）有关。提议者将研究具有边界的表面的这种极端配置。调查的另一个主要领域涉及一般相对论的爱因斯坦方程。这些方程式描述了宇宙中大体体的重力场。该理论纯粹是几何，是具有初始值表述的波理论。提议者计划研究解决方案的几何形状，以给出发生重力崩溃并形成黑洞的条件。此类问题导致重要的几何问题，涉及引力能和时空的曲率。拟议的研究是在差分几何，一般相对论和部分微分方程之间的接口处。几何研究的主要主题将是光谱几何形状的研究。 PI计划在表面上构建指标，并受到最大化第一个特征值的面积或边界长度约束的较高较高的歧管。这是一种非标准的变异问题类型，因为它涉及在竞争者的无限维空间上最大化和最小化。 PI将研究这种最大化指标的几何形状，以确定最佳基本频率的最佳形状。相对论，PI打算继续对Bartnik提出的结构进行调查，以最大程度地减少紧凑型域和静态真空度量的扩展。 PI还打算研究有关它们是否可以包含非紧凑型稳定捕获表面的问题的初始数据集的几何特性。最后，PI打算研究约束方程解的模量空间的全局性能，以定义爱因斯坦方程的可能初始数据。 PI应提出一系列有关满足自由边界条件以及与特征值问题的联系的最小子曼群的问题。最终，PI计划继续研究Kahler-Einstein歧管的最小拉格朗日和特殊的拉格朗日submanifolds。这项研究将试图证明有关卡拉比yau歧管的整体同源性不变性的猜想，当一个人变形时，该歧管的整体同源性是由最小拉格朗日循环产生的。