Geometric analysis problems related to surfaces in mathematical physics

数学物理中与曲面相关的几何分析问题

• 批准号: 0904281
• 负责人: Mu-Tao Wang
• 金额: $ 13万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0904281&HistoricalAwards=false
• 关键词: Geometric; analysis; problems; related; surfaces

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.

王慕涛教授计划研究数学物理中的曲面几何及其相关的偏微分方程组。他计划继续研究Kahler-Einstein流形的拉格朗日子流形的平均曲率流。直接的目标是理解在自然凸性条件下流动的行为，并将结果应用于同位素问题。王教授将与邱成栋教授一起进一步研究他最近发现的准局部质量的概念。近期目标包括研究准局域质量在空间、零和类时间方向上的变化，以及计算爱因斯坦场方程解上的准局域能量动量矢量。王慕涛教授的目的是研究与特殊弯曲几何形状相关的几何和分析，包括作为肥皂膜的高维类似的特殊拉格朗日，以及作为空间区域边界的时空表面。特殊拉格朗日的研究将为Calabi-Yau流形，弦理论空间的几何提供新的视角和见解。王教授新发现的准局域能量衡量的是宇宙有界区域中包含的总能量，即使边界上的引力效应非常强。这个概念在许多涉及广义相对论奇点的悬而未决的问题中是必不可少的。这项拟议中的研究将进一步加深我们对宇宙中的能量以及黑洞形成和碰撞等相关问题的理解。