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教授建议在微分几何和广义相对论领域进行研究。他建议研究解决方案的约束方程是不是时间对称的眼睛的研究一般彭罗斯不等式以及分析的稳定性的约束流形。后者的主题是重要的理解数值稳定性的真空爱因斯坦方程。Schoen教授还建议研究由特殊的拉格朗日校准形式校准的子流形的构造。他计划将几何测量理论的思想应用于这个问题。他还计划研究维度大于二的汉密尔顿平稳子流形;特别是，将研究汉密尔顿平稳切锥。Schoen教授打算研究在覆盖下保持稳定的稳定极小曲面，希望表明这些曲面在一般情况下是全纯的。最后，他打算研究常高斯-克罗内克曲率的闵可夫斯基空间中超曲面的测地完备性。舍恩教授的项目将使人们更好地理解爱因斯坦广义相对论方程的解。更好的理论理解是成功的精确数值模拟的解决方案。数值模拟对于预测动态情况下产生的引力辐射的性质非常重要，NSF目前有一个大型项目LIGO，试图测量这种辐射。他认为，这个项目的理论工作将有助于数字。Schoen教授提出的工作的其余部分涉及使用几何方法来理解表面界面的行为，如肥皂膜和肥皂泡，在物理情况下出现的不同尺寸。这些自然的几何物体可以用来描述它们所处的空间的微妙特性。这些空间出现在物理模型中，例如弦理论，在那里它们扮演着基本的角色。