Differential Geometry and Partial Differential Equations

DMS-0104163PI摘要：理查德·M·舍恩教授提议研究广义相对论时空中定义区域的Bartnik准局部质量的极值度量的存在性。这将产生静态真空爱因斯坦方程在适当的边界条件下的渐近平坦解的存在性定理。这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在Calabi-Yau流形和Kaehler-Estein流形中构造特殊的拉格朗日和更一般的极小拉格朗日子流形。方法是通过拉格朗日子流形之间的直接体积最小化来构造哈密顿平稳子流形，并得到充分的正则性来证明它们是极小拉格朗日的。Schoen的第三个项目是进一步发展调和映射方法来证明紧流形上半单Lie群中格的光滑作用的刚性。王慕涛教授建议研究余维大于1的特殊子流形的平均曲率流。其主要目的是获得流动的稳定性和规律性。杨宝章博士将研究杨-Mills连接在任意维上的奇异行为。这项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。这项研究项目涉及几何形状的研究，它优化了某些物理和几何能量。对于广义相对论中的弯曲时空，没有可以分配给引力场的自然质量-能量密度，因此巴特尼克建议通过最小化包含一个区域作为子集的所有物理时空的总质量来测量该区域在一个时空中的引力质量。这个最小质量时空，如果能证明它的存在，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构造这种静态解的方法，并将它们用于研究三维几何。人们期望三维空间上有自然的几何形状，这些几何形状由它们的曲率性质来唯一地表征。这个项目的另一个目标是在弦理论的空间Calabi-Yau流形中构造某些特殊的曲面。这些(三维)表面被称为特殊拉格朗日子流形，它们类似于肥皂膜，因为它们是面积最小的表面。最后，提出了研究曲面的平均曲率演化问题。这是一个演化问题，它以曲面在空间中移动的方式使其面积最快地减小。了解这种演化的行为对于以最佳方式简化和平滑复杂曲面非常重要。从数学上讲，这是一个困难的问题，因为表面可能会产生奇点，如圆锥点和撕裂，这必须被考虑在内。

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.