Differential Geometry and Partial Differential Equations

计划研究的一个方面与表面的最佳形状有关。如果我们把表面看作一个以一定频率自由振动的鼓面，那么，粗略地说，几何形状越复杂，它的基频就越小。 这就提出了寻找使其面积的基频最大化的几何形状的问题。这个极值问题是一个困难的和许多研究的问题。事实证明，出现的几何形状与面积最小的表面（肥皂膜）有关。首席研究员将研究这种极端配置的表面与边界。另一个主要的研究领域是广义相对论的爱因斯坦方程。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，并且是具有初始值公式的波动理论。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。这样的问题导致重要的几何问题，包括引力能和时空曲率。 该研究是在微分几何，广义相对论和偏微分方程之间的接口。几何研究的一个主题将是谱几何的研究。 一个项目的目的是构造度量表面和某些高维流形的面积或边界长度的限制，最大化的第一个特征值。这是一个非标准类型的变分问题，因为它涉及到最大化和最小化无限维空间的竞争对手。 首席研究员将研究这种最大化度量的几何形状，以确定具有最大基频的最佳形状。在相对论方面的工作将继续调查的建设提出的Bartnik的质量最小化扩展紧凑域和静态真空度规。 首席研究员还打算研究初始数据集的几何特性，以解决它们是否可以包含非紧凑稳定的捕获表面的问题。首席研究员打算研究约束方程解的模空间的全局性质，这些约束方程定义了爱因斯坦方程可能的初始数据。一系列的问题将追求有关极小子流形满足自由边界条件和连接到特征值问题。在继续研究的最小拉格朗日和特殊拉格朗日子流形的凯勒-爱因斯坦流形的主要研究者将试图证明一个猜想有关的子群的不变性的积分同调的卡-丘流形这是由最小拉格朗日周期时产生的变形环境卡-丘结构。

One aspect of the planned 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 will be its fundamental frequencies. 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 principal investigator 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. One project aims 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 principal investigator will study the geometry of such maximizing metrics to determine the optimal shapes with largest fundamental frequency. Work in relativity will continue investigations into the construction proposed by Bartnik of mass minimizing extensions of compact domains and static vacuum metrics. The principal investigator also intends to study geometric properties of initial data sets to address the question of whether they can contain non-compact stable trapped surfaces. The principal investigator 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. A range of questions will be pursued concerning minimal submanifolds satisfying free boundary conditions and connections to eigenvalue problems. In a continuing study of minimal Lagrangian and special Lagrangian submanifolds of Kaehler-Einstein manifolds the principal investigator 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.