Differential Geometry and Partial Differential Equations

DMS-0104163PI的摘要：Richard M. Schoen Schoen教授提议研究广义相对论中一个域的Bartnik准局域质量的极值度规的存在性。这将得到具有适当边界条件的静态真空爱因斯坦方程的渐近平坦解的存在性定理。它是构造三维几何的变分方法的一部分。Schoen的第二个项目涉及到Calabi-Yau和kaehler - einstein流形中的特殊拉格朗日和更一般的最小拉格朗日子流形的构造。该方法是利用拉格朗日子流形之间的直接体积最小化构造哈密顿平稳子流形，并获得足够的正则性来证明它们是最小拉格朗日流形。Schoen的第三个项目是进一步发展调和映射方法，以证明紧流形上半单利群中格的光滑作用的刚性。王木涛教授提出研究余维数大于1的特殊类型子流形的平均曲率流。主要推力是获得流动的稳定性和规律性。杨宝章博士将研究任意维杨-米尔斯连接的奇异行为。本研究将包括对爆破集的结构和奇点附近的渐近行为的研究。这个研究项目涉及几何形状的研究，使某些物理和几何能量最优化。对于弯曲时空广义相对论来说，没有可以分配给引力场的自然质量-能量密度，因此巴特尼克提出通过最小化包含该区域作为子集的所有物理时空的总质量来测量时空中一个区域的引力质量。这个最小质量时空，如果能被证明存在的话，将是爱因斯坦方程的静态解。这个项目的目标之一是找到一种方法来构建这样的静态解，并用它们来研究三维几何。人们期望三维空间具有具有独特曲率特性的自然几何。本项目的另一个目标是在弦论空间Calabi-Yau流形中构造某些特殊曲面。这些（三维）表面被称为特殊拉格朗日子流形，类似于肥皂膜，因为它们是面积最小的表面。最后，提出研究曲面的演化问题，即平均曲率演化问题。这是一个进化问题，它在空间中移动一个表面，使其面积减少得最快。了解这种演化的行为对于以最优方式简化和平滑复杂表面具有重要意义。这在数学上是一个难题，因为表面可能会产生奇点，如锥点和撕裂，这些都必须考虑在内。

Abstract for DMS-0104163PI: Richard M. SchoenProfessor Schoen is proposing to study the existence of an extremal metricwhich defines the Bartnik quasilocal mass for a domain in a spacetime ofgeneral relativity. This would yield an existence theorem forasymptotically flat solutions of the static vacuum Einstein equations withsuitable boundary conditions. It is part of a variational approach forconstructing three dimensional geometries. Schoen's second projectinvolves the construction of special lagrangian, and more generallyminimal lagrangian, submanifolds in Calabi-Yau and Kaehler-Einsteinmanifolds. The approach is to construct hamiltonian stationarysubmanifolds by direct volume minimization among lagrangian submanifolds,and to obtain sufficient regularity to show that they are minimallagrangian. Schoen's third project is to further develop the harmonic mapapproach to prove the rigidity of smooth actions of lattices in semisimpleLie groups on compact manifolds. Professor Mutao Wang proposes to studythe mean curvature flow for special classes of submanifolds ofcodimension greater than one. The major thrust is to obtain stability andregularity properties of the flow. Dr. Baozhang Yang will study singularbehavior of Yang-Mills connections in arbitrary dimension. This study willinclude an investigation into the structure of blow-up sets and asymptoticbehavior near singularities. This research project concerns the study of geometric shapes whichoptimize certain physical and geometric energies. For curved spacetimes ingeneral relativity, there is no natural mass-energy density which can beassigned to the gravitational field, so Bartnik proposed to measure thegravitational mass of a region in a spacetime by minimizing the total massof all physical spacetimes which contain this region as a subset. Thisminimal mass spacetime, if it can be shown to exist, will be a staticsolution of Einstein's equations. One of the goals of this project is tofind a way to construct such static solutions, and to use them to studythree dimensional geometry. It is expected that three dimensional spaceshave natural geometries on them which are uniquely characterized by theircurvature properties. Another goal of this project is to construct certainspecial surfaces in Calabi-Yau manifolds, the spaces of stringtheory. These (three dimensional) surfaces, called special lagrangiansubmanifolds, are analogous to soap films in that they are surfaces of least possible area. Finally, it is proposed to study the evolutionproblem for surfaces which is called the mean curvature evolution. This isan evolution problem which moves a surface in space in such a way that itsarea is decreased most rapidly. Understanding the behavior of thisevolution is important for simplifying and smoothing complicated surfacesin an optimal way. Mathematically this is a difficult problem because thesurfaces may develop singularities such as cone points and tears whichmust be accounted for.