Differential Geometry and Partial Differential Equations

摘要提案：DMS-9803341首席研究员：Richard M. Schoen该提案涉及三个几何变分问题。首先是辛流形的拉格朗日子流形的体积最小化问题。该理论提供了一种构造Kahler-Einstein流形的特殊拉格朗日子流形的方法。第二个问题是关于离散群在非正曲率空间上的一般等距作用的等变调和映射的研究。该理论可用于研究离散群有限维和无限维表示的许多一般刚性问题。第三个问题涉及爱因斯坦度量的变分问题，其中的问题是更一般地计算Yamabe不变量，并表明标准度量达到最小-最大变分特征。最小化问题出现在数学和科学的许多分支中。例如，线性规划涉及到一组约束不等式下最小化函数（如成本）的问题，导航问题涉及到在地球表面上寻找最小长度的路径，而在连续介质力学中，弹性膜的平衡位置是通过势能尽可能小的条件在无限多个可能的位置中确定的。本文提出了一类几何变分问题，其中第一个问题是在映射保持面积的约束下，映射势能的最小化问题。这种问题出现在非线性弹性中，其中映射表示弹性体的变形。问题也出现在几何中，人们可以使用这种最小化配置来理解弦理论（物理）中出现的复杂几何空间。该建议的第二部分处理在复杂对称群的对称条件下最小化势能的映射。例如，如果你考虑围绕给定区域的最小长度曲线，你会得到一个圆，而如果你选择围绕其平移（在固定对称群下）填充平面的区域的曲线，那么解决方案通常是一种特殊类型的六边形（如果对称群选择合适，蜂巢的正六边形）。主要目标是使用对称最小化映射来理解可能的对称群，以及它们如何在重要的几何情况下出现。建议的最后一部分处理爱因斯坦广义相对论方程的平衡解。完整的爱因斯坦方程可以被认为是描述引力场的振动，引力场决定了时空的几何形状。相应的平衡问题在几何中是重要的，建议的第三部分处理平衡解在多大程度上可以使势能最小化的问题。

Abstract Proposal: DMS-9803341 Principal Investigator: Richard M. Schoen This proposal deals with three geometric variational problems. The first is the problem of minimizing the volume for lagrangian submanifolds of symplectic manifolds. This theory provides an approach to constructing special lagrangian submanifolds of Kahler-Einstein manifolds. The second problem is the study of harmonic maps which are equivariant with respect to general isometric actions of discrete groups on spaces of nonpositive curvature. This theory may be applied to study many general rigidity questions for both finite and infinite dimensional representations of discrete groups. The third problem concerns the variational problem for Einstein metrics, where the problem is to compute the Yamabe invariant in more generality, and to show that standard metrics achieve this min-max variational characterization. Minimization problems occur in many branches of mathematics and science. For example, linear programming concerns the problem of minimizing a function (such as cost) subject to a set of constraint inequalities, problems of navigation involve finding paths of least length on the earth's surface, and in continuum mechanics, the equilibrium position of an elastic membrane is determined among the infinitely many possible positions by the condition that the potential energy be as small as possible. This proposal deals with certain geometric variational problems, of which the first is the problem of minimizing a potential energy for mappings subject to the constraint that the mappings preserve the area. Such problems arise in nonlinear elasticity where the mapping represents the deformation of an elastic body. The problems also arise in geometry where one can use such minimizing configurations to understand complicated geometric spaces that arise in string theory (physics). The second part of this proposal deals with maps which minimize a potential energy subject to the conditi on that they are symmetric for a complicated symmetry group. For example, if you consider the curve of least length which surrounds a given area, you get a circle, while if you choose curves which surround regions whose translates (under a fixed symmetry group) fill up the plane, then the solution is typically a special type of hexagon (the regular hexagon of the honeycomb if the symmetry group is chosen suitably). The main goal is to use the symmetric minimizing maps to understand the possible symmetry groups and how they can arise in important geometric situations. The final part of the proposal deals with equilibrium solutions of the Einstein equations of General Relativity. The full Einstein equations may be thought of as describing the vibrations of the gravitational field which determines the geometry of spacetime. The corresponding equilibrium problem is important in geometry, and the third part of the proposal deals with the question of the extent to which equilibrium solutions can be expected to minimize the potential energy.