Applications of geometric analysis to general relativity and geometric flows

AbstractAward：DMS 1405152，首席研究员：王慕涛首席研究员建议研究广义相对论中的质量、能量、角动量和质心的概念。这些概念在物理学的任何分支中都是极其重要的。然而，自爱因斯坦时代以来，很难找到物理上可接受的引力概念的定义。许多尚未解决的问题的解决，如黑洞如何形成和黑洞如何碰撞，基本上依赖于这些概念。最近，首席研究员和他的合作者成功地发现了孤立系统（例如，当观察者离星星很远时）和非孤立系统（例如，当观察者在近距离内，两颗恒星相互旋转时）的定义。他们发现的定义满足许多非常理想的性质，包括爱因斯坦方程的第一个精确的动力学描述。这项提议描述了进一步探索这些新定义及其应用的计划。预计该项目所取得的成果将成为更深入地了解大规模宇宙的关键步骤。 首席研究员还建议研究几何流动。这些微分方程对几何形状如何变形并以最有效的方式演变为最佳形式进行建模。这项研究有望在广义相对论和数学物理学中得到应用。 另一项研究将研究准局部质量和准局部守恒量的应用，如角动量和质心，他最近与合作者发现，旨在锚和解决经典广义相对论中的几个问题。 这个提议的近期目标包括解决广义相对论中的不变质量猜想，在准局部和总体水平上证明守恒量的定义，以及在爱因斯坦方程动力学研究中的应用。首席研究员还将继续他的研究反平均曲率流和平均曲率流。 近期目标包括证明史瓦西时空中的Gibbons-Penrose不等式和平均曲率流的动力学稳定性。

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.