Differential Geometry and Partial Differential Equations

所提出的研究的一个方面与表面的最佳形状有关。如果我们把表面想象成一个在一定频率下自由振动的鼓面，那么，粗略地说，几何形状越复杂，它的基频就越小。这就提出了寻找几何形状的问题，使其面积的基频最大化。这个极端的问题是一个困难的、研究得很多的问题。结果表明，产生的几何形状与最小面积表面（肥皂膜）有关。本文将研究具有边界的曲面的这种极值构型。另一个主要研究领域涉及爱因斯坦广义相对论方程。这些方程描述了宇宙中大质量物体的引力场。该理论是纯几何的，是一个具有初值公式的波动理论。这位提议者计划研究解的几何形状，以给出引力坍缩发生和黑洞形成的条件。这些问题引出了一些重要的几何问题，包括引力能和时空曲率。提出的研究是在微分几何，广义相对论和偏微分方程之间的接口。几何研究的一个主要主题将是光谱几何的研究。PI计划在曲面和某些高维流形上构造度量，这些流形受面积或边界长度约束，使第一个特征值最大化。这是一个非标准类型的变分问题，因为它涉及竞争对手的无限维空间的最大化和最小化。PI将研究这些最大化度量的几何形状，以确定具有最大基频的最佳形状。在相对论中，PI打算继续研究Bartnik提出的紧域的质量最小化扩展和静态真空度量的构造。PI还打算研究初始数据集的几何性质，涉及它们是否可以包含非紧致稳定捕获表面的问题。最后，研究了约束方程解的模空间的整体性质，这些约束方程定义了爱因斯坦方程可能的初始数据。PI应该研究一系列关于满足自由边界条件的最小子流形以及与特征值问题的联系的问题。最后，PI计划继续他的最小拉格朗日和特殊拉格朗日子流形的研究。本文试图证明一个关于Calabi-Yau流形的积分同调子群的不变性的猜想，该流形是由最小拉格朗日循环产生的，当周围的Calabi-Yau结构变形时。

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.