Geometric analysis problems related to surfaces in mathematical physics

王慕涛教授提出研究数学物理中的曲面几何及其相关的PDE。他计划继续他对卡勒-爱因斯坦流形的拉格朗日子流形的平均曲率流的研究。当前的目标是了解自然凸性条件下的流动行为，并将结果应用于同位素问题。王教授将与丘成东教授进一步探讨他最近发现的准局部质量的概念。近期目标包括研究准局部质量如何在空间、零和类时方向上变化，以及评估爱因斯坦场方程解上的准局部能量动量向量。王慕涛教授主要研究与特殊弯曲几何形状相关的几何和分析，包括与肥皂膜类似的高维特殊拉格朗日量，以及作为空间区域边界的时空表面。特殊拉格朗日量的研究将为弦理论空间中Calabi-Yau流形的几何提供新的视角和见解。王教授新发现的准局域能量测量了宇宙中一个有界区域所包含的总能量，即使边界上的引力作用非常强。这个概念在广义相对论中许多涉及奇点的未解决问题中是必不可少的。这项提议的研究将进一步加深我们对宇宙能量以及黑洞形成和碰撞等相关问题的理解。

Professor Mu-Tao Wang proposes to study the geometry of surfaces in mathematical physics and their related PDE's. He plans to continue his research on the mean curvature flow of Lagrangian submanifolds of Kahler-Einstein manifolds. Immediate goals are to understand the behavior of the flow under natural convexity conditions and to apply the results to isotopy problems. Professor Wang shall further investigate the notion of quasilocal mass he recently discovered with Professor Shing-Tung Yau. Immediate goals include studying how the quasi-local mass changes in the spatial, null, and timelike directions and evaluating the quasi-local energy-momentum vector on solutions of the Einstein field equation.Professor Mu-Tao Wang purposes to study the geometry and analysis associated with special curved geometric shapes, including special Lagrangians which are higher dimensional analogues of soap films, and surfaces in space-time that are boundaries of spatial regions.The study of special Lagrangians will provide new perspectives and insights of the geometry of Calabi-Yau manifolds, the space of string theory. Professor Wang's newly discovered quasi-local energy measures the total energy contained in a bounded region of the universe even when the gravitational effect on the boundary is very strong.This notion is essential in many unsolved problems involving singularities in general relativity. The proposed research will further our understanding of energy in the universe and related problems such as black hole formations and collisions.