Problems in General Relativity and Geometric Flows

这个项目涉及在广义相对论、几何和微分方程界面上的基本问题的研究。爱因斯坦的广义相对论描述了时空是如何被引力弯曲的。他的理论语言是几何学，这种现象是由他的同名方程控制的。广义相对论在理论和实验方面都取得了巨大的进步，例如最近LIGO探测到的引力波。然而，由于爱因斯坦方程的复杂性，我们对宇宙的大部分认识都是全局的和大尺度的，比如从非常遥远的距离观察一个天体物理事件。最近，PI和他的合作者运用几何和微分方程的工具，对宇宙中任何有限扩展区域的引力能和质量进行了最精确的测量。这对于理解我们宇宙的精细和局部结构是至关重要的，例如在GPS技术和空间探索中的应用。它在研究非孤立的大规模现象，如黑洞合并和碰撞方面也至关重要。该项目的成功将加深我们对引力能/质量和时空的非线性局域/全局性质的理解。PI还计划研究几何流，这是一种微分方程，它模拟几何形状如何以最有效的方式变形并演变成最佳形式。PI一直致力于培养多样化的研究生和年轻研究人员，该项目将有助于他继续朝着这个方向努力。PI计划用几何分析的方法解决广义相对论和几何流中的几个突出问题。特别是，PI与Yau一起发现的准局部质量定义将是本研究的关键因素。近期目标包括证明一般不变质量猜想，建立Robinson-Trautman时空的正质量定理和高维史瓦西时空的线性稳定性，以及用拉格朗日平均曲率流求解Arnol猜想。该研究项目也将促进我们对非线性偏微分系统的理解，如最优等距嵌入方程、平均曲率流和爱因斯坦方程。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project concerns investigations of fundamental problems at the interface of general relativity, geometry, and differential equations. Einstein's theory of general relativity describes how the spacetime is curved by gravitation. The language of his theory is geometry and the phenomenon is governed by his eponymous equation. There have been great advances in both theoretical and experimental aspects of general relativity, for example the recent detection of gravitational waves by LIGO. However, due to the complexity of Einstein's equation, most of our knowledge of the universe are global and large scale, such as the observation of an astrophysical event from a very faraway distance. Recently, the PI and his collaborators applied the tools of geometry and differential equations to give the most precise measurement of gravitational energy and mass on any finitely extended region of the universe. This is essential in understanding the fine and local structure of our universe, with applications in, for example, GPS technology and space exploration. It is also crucial in studying non-isolated large-scale phenomena such as black hole mergers and collisions. The success of this project will deepen our understanding of gravitational energy/mass and the nonlinear local/global nature of the spacetime. The PI also plans to study geometric flows, which are differential equations that model how a geometric shape deforms and evolves to an optimal form in the most efficient way. The PI has been engaging himself in educating a diversified body of graduate students and young researchers, and the project will be instrumental for his continued efforts along this direction.The PI plans to resolve several outstanding problems in general relativity and geometric flows by the method of geometric analysis. In particular, the quasilocal mass definition the PI discovered with Yau will be a key ingredient in this research. Immediate goals include proving the general invariant mass conjecture, establishing a positive mass theorem for the Robinson-Trautman spacetime and the linear stability of higher dimensional Schwarzschild spacetimes, and the solution of a conjecture of Arnol'd by the Lagrangian mean curvature flow. This research project will also advance our understanding of nonlinear partial differential systems such as the optimal isometric embedding equation, the mean curvature flow, and the Einstein equation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.