Problems in General Relativity and Geometric Flows

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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.