Differential Geometry and Partial Differential Equations

微分几何和偏微分方程

• 批准号: 9803341
• 负责人: Richard Schoen
• 金额: $ 22.21万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803341&HistoricalAwards=false
• 关键词: Differential; Geometry; Partial; Equations

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.

摘要 提案：DMS-9803341主要研究者：Richard M. Schoen 这个建议涉及三个几何变分问题。 第一个问题是辛流形的拉格朗日子流形的体积极小化问题。 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。 第二个问题是研究非正曲率空间上关于离散群的一般等距作用等变的调和映射。 这一理论可用于研究离散群的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，这里的问题是计算Yamabe不变量的更一般性，并表明标准度量实现这一最小-最大变分表征。 极小化问题存在于数学和科学的许多分支中。 例如，线性规划关注的是在一组约束不等式下最小化函数（如成本）的问题，导航问题涉及在地球表面上找到最短长度的路径，在连续介质力学中，弹性膜的平衡位置是在无限多个可能的位置中确定的，条件是势能尽可能小。 这个建议处理某些几何变分问题，其中第一个问题是最小化的势能映射的约束下，映射保持面积。 这样的问题出现在非线性弹性力学中，其中映射表示弹性体的变形。 这个问题也出现在几何学中，人们可以使用这种最小化配置来理解弦理论（物理学）中出现的复杂几何空间。 这个建议的第二部分涉及的地图，最小化的势能受到的条件，他们是对称的一个复杂的对称群。 例如，如果你考虑围绕给定区域的最短长度的曲线，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（如果对称群选择适当，则是蜂窝的正六边形）。 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。 该提案的最后一部分涉及广义相对论爱因斯坦方程的平衡解。 完整的爱因斯坦方程可以被认为是描述了决定时空几何的引力场的振动。 相应的平衡问题是重要的几何，和第三部分的建议处理的问题，在何种程度上的平衡解决方案可以预期尽量减少势能。