Problems in general relativity and geometric flows

王慕涛教授提出用几何分析的方法来研究广义相对论和几何流中的几个问题。他将进一步研究他最近与邱成东发现的准局部质量和能量动量的性质。直接目标包括求解最优等距嵌入方程，理解准局部质量的单调性和亚加性，以及锚定准局部角动量的定义。王教授计划继续研究拉格朗日子流形的平均曲率流。近期目标包括证明两种情况下流的整体存在性和收敛性(1)Kaehler-Einstein流形的辛形态图(2)协切束的精确拉格朗日子流形，并将这些结果应用于拉格朗日同位素问题。由于引力能的非局域性，ADM或邦迪质量只有在引力较弱的无穷远处才有其表达。然而，大多数可观测的物理模型都是有限扩展的区域，在这样一个区域上测量质量/能量对于广义相对论的许多基本问题是必不可少的。王教授新发现的准局部能量评估了宇宙中任何有界区域所包含的总能量，即使引力的影响非常强。提出的研究将进一步加深我们对大尺度引力能和与黑洞有关的问题的理解。王教授的目的是研究曲面的“平均曲率流”。这种流动就像热方程，它耗散曲率分布并使表面变形为平衡状态。他计划运用这一流程将普通表面与具有最佳形状的表面连接起来。这个过程将帮助我们理解这些表面的几何和拓扑结构。它在计算机图形学和图像处理等应用领域也很重要。

Professor Mu-Tao Wang proposes to apply the method of geometric analysis to study several problems in general relativity (GR) and geometric flows. He will further investigate properties of quasilocal mass and energy-momentum which he recently discovered with Shing-Tung Yau. Immediate goals include solving the optimal isometric embedding equation, understanding monotone and sub-additive properties of the quasilocal mass, and anchoring the definition of quasilocal angular momentum. Professor Wang plans to continue his research on the mean curvature flow of Lagrangian submanifolds. Immediate goals include proving the global existence and convergence of the flow in two cases (1) graphs of symplectomorphisms of Kaehler-Einstein manifolds (2) exact Lagrangian submanifolds of cotangent bundles, and applying these results to Lagrangian isotopy problems.Due to the non-local feature of gravitational energy, the ADM or Bondi mass only finds its expression at infinity where gravitation is weak. However, most observable physical models are finitely extended regions and measurement of mass/energy on such a region is essential in many fundamental issues in GR. Professor Wang's newly discovered quasi-local energy evaluates the total energy contained in any bounded region of the universe even if the effect of gravitation is very strong. The proposed research will further our understanding of gravitational energy in large scale and of problems related to black holes. Professor Wang purposes to study the ``mean curvature flow" of surfaces. This flow is like the heat equation which dissipates curvature distributions and deforms a surface into an equilibrium state. He plans to apply this flow to connect a general surface to ones with optimal shape. This procedure will help us understand the geometry and topology of such surfaces. It is also important in applied areas such as computer graphics and image processing.