Differential Geometry and Partial Differential Equations

微分几何和偏微分方程

• 批准号: 0604960
• 负责人: Richard Schoen
• 金额: $ 67.66万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604960&HistoricalAwards=false
• 关键词: Differential; Geometry; Partial; Equations

Professor Schoen is proposing research in the areas of Differential Geometry and General Relativity. He is proposing to study solutions of the constraint equations which are not time symmetric with an eye to the study of the general Penrose inequality as well as an analysis of the stability of the constraint manifold. The latter topic is important for understanding numerical stability for the vacuum Einstein equations. Professor Schoen is also proposing to study the construction of submanifolds which are calibrated by the special lagrangian calibrating form. He plans to apply ideas from geometric measure theory to this problem. He also plans to study hamiltonian stationary submanifolds in dimension greater than two; in particular, the hamiltonian stationary tangent cones will be studied. Professor Schoen intends to investigate stable minimal surfaces which remain stable under coverings with the hope of showing that these are holomorphic in general situations. Finally, he intends to study geodesic completeness properties of hypersurfaces in Minkowski space of constant Gauss-Kronecker curvature. Professor Schoen's project will lead to a better understanding of solutions of the Einstein equations of General Relativity. A better theoretical understanding is essential for the success of accurate numerical modeling of solutions. Numerical modeling is important for predicting the nature of the gravitational radiation which arises from dynamic situations, and NSF currently has a large project, LIGO, which is attempting to measure this radiation. He believes that the theoretical work of this project will be helpful for the numerics. The remainder of Professor Schoen's proposed work involves the use of geometric methods to understand the behavior of surface interfaces, such as soap films and soap bubbles, of varying dimension which arise in physical situations. These natural geometric objects can be used to describe subtle properties of the spaces in which they reside. These spaces arise in physical models such as string theory where they play a basic role.

Schoen教授正在提出在差异几何和一般相对论领域的研究。他提议研究约束方程的解决方案，这些方程不是对称时间对称的，以研究一般的penrose不平等的研究以及对约束歧管的稳定性的分析。后一个主题对于理解真空爱因斯坦方程的数值稳定性很重要。 Schoen教授还建议研究通过特殊的Lagrangian校准形式校准的子曼菲尔德的构建。他计划将几何测量理论的想法应用于这个问题。他还计划在大于两个的维度上学习哈密顿固定的子曼佛。特别是，将研究哈密顿固定的切线锥。 Schoen教授打算调查稳定的最小表面，这些表面在覆盖范围内保持稳定，希望表明这些表面在一般情况下都是霍顿。最后，他打算研究恒定高斯 - 克罗内克曲率的Minkowski空间中的高曲面的大地完整性。 Schoen教授的项目将使人们对爱因斯坦方程的解决方案有更好的了解。更好的理论理解对于成功的准确数值建模至关重要。数值建模对于预测由动态情况产生的重力辐射的性质很重要，而NSF当前具有大型项目，该项目正在试图测量这种辐射。他认为，该项目的理论工作将对数字有帮助。 Schoen教授提出的工作的其余部分涉及使用几何方法来了解在物理情况下出现的不同尺寸的表面界面的行为，例如肥皂膜和肥皂泡。这些天然的几何对象可用于描述其居住空间的微妙特性。这些空间出现在物理模型中，例如弦理论，它们扮演基本角色。