Applications of geometric analysis to general relativity and geometric flows

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AbstractAward: DMS 1405152, Principal Investigator: Mu-Tao WangThe principal investigator proposes to study the notions of mass, energy, angular momentum, and center of mass in general relativity. These notions are of most fundamental importance in any branch of physics. However, since Einstein's time, there has been great difficulty to find physically acceptable definitions of these concepts for gravitation. The solutions of many unsolved problems such as how a black hole forms and how black holes collide rely essentially on these notions. Recently, the principal investigator and his collaborators successfully discovered definitions for both isolated systems (e.g. when the observer is very far away from a star) and non-isolated systems (e.g. when the observer is at close range with two stars rotating about each other ). The definitions they found satisfy many highly desirable properties, including the first precise dynamical description of Einstein's equation.This proposal describe plans to further explore these new definitions and their applications. The results obtained in this project are expected to be crucial steps towards deeper understanding of the universe in a large scale. The principal investigator also proposes to study geometric flows. These are differential equations that model how a geometric shape deforms and evolves to an optimal form in the most efficient way. The proposed research is expected to have applications in general relativity and mathematical physics. Another line of investigation will study applications of quasi-local mass and quasi-local conserved quantities such as angular momentum and center of mass, which he recently discovered with his collaborators, aiming to anchor and resolve several problems in classical general relativity. Immediate goals of this proposal include resolving the invariant mass conjecture in general relativity, justification of the definitions of conserved quantities at both the quasi-local and total levels, and applications in the study of the dynamics of the Einstein equation. The principal investigator will also continue his research on inverse mean curvature flows and mean curvature flows. Immediate goals include the proof of a Gibbons-Penrose inequality in Schwarzschild spacetime and the dynamical stability of the mean curvature flow.