Geometric Partial Differential Equations in General Relativity

广义相对论中的几何偏微分方程

• 批准号: 1308837
• 负责人: Lan-Hsuan Huang
• 金额: $ 28.22万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308837&HistoricalAwards=false
• 关键词: Geometric; Partial; Differential; Equations; General

The goal of this proposal is to understand the Einstein constraint equations and to explore new geometric properties of asymptotically flat manifolds. The first part of the project continues the studies of the PI and co-PI on hypersurfaces with nonnegative scalar curvature. They propose to classify scalar-flat hypersurfaces and to apply the arguments to study the ADM mass inequalities. The second part of the project studies the density theorems to the full Einstein constraint equations and their relations to the global conserved quantities, including center of mass and angular momentum. The proposed research will also have applications to the complex Hessian equations which generalize the Calabi-Yau equations on Kaehler manifolds.The project studies several fundamental problems to better understand the analytical and geometric structure of the mathematical models of our universe. The proposed techniques connect different branches of mathematics and physics, including general relativity, Riemannian geometry, complex geometry, and partial differential equations. The new ideas have the potential for setting long-standing questions in classical differential geometry. Moreover, the proposal will generate research problems suitable for undergraduate and graduate students and lead to research activities in the New England area to attract more students into the current active research fields.

这个提议的目标是理解爱因斯坦约束方程和探索渐近平坦流形的新几何性质。该项目的第一部分继续研究具有非负纯量曲率的超曲面上的PI和co-PI。他们提出分类标量平坦超曲面和应用的参数来研究ADM质量不等式。第二部分研究完整的爱因斯坦约束方程的密度定理及其与整体守恒量的关系，包括质心和角动量。该研究还将应用于Kaehler流形上的Calabi-Yau方程的复杂Hessian方程。该项目研究了几个基本问题，以更好地理解我们宇宙数学模型的分析和几何结构。所提出的技术连接了数学和物理的不同分支，包括广义相对论，黎曼几何，复几何和偏微分方程。新的想法有可能设置长期存在的问题，在经典微分几何。此外，该提案将产生适合本科生和研究生的研究问题，并引发新英格兰地区的研究活动，以吸引更多学生进入当前活跃的研究领域。