Differential Geometry and Partial Differential Equations

DMS-0104163 PI摘要：Richard M. Schoen教授提议研究一个极端度规的存在，它定义了广义相对论时空中一个域的Bartnik准局部质量。这将得到一个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的第二个项目涉及建设的特殊拉格朗日，更一般的最小拉格朗日，子流形在Calabi-Yau和Kaehler-Einsteinmanifold。其方法是在拉格朗日子流形中直接通过体积极小化来构造哈密顿定常子流形，并得到充分的正则性来证明它们是极小拉格朗日子流形。Schoen的第三个项目是进一步发展调和映射的方法来证明刚性的顺利行动格在semisimpleLie群紧凑的流形。王慕涛教授提出研究余维大于1的特殊子流形类的平均曲率流。主要的目的是获得流动的稳定性和规律性.杨宝章博士将研究任意维Yang-Mills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。这个研究项目关注的是优化某些物理和几何能量的几何形状的研究。对于广义相对论中的弯曲时空，没有自然的质能密度可以分配给引力场，因此Bartnik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某个区域的引力质量。这个最小质量的时空，如果能够被证明存在的话，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决方案，并使用它们来研究三维几何。人们期望三维空间具有自然的几何形状，这些几何形状具有独特的曲率性质。这个项目的另一个目标是在卡-丘流形（弦论空间）中构造某些特殊曲面。这些（三维）表面，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，称之为平均曲率演化。这是一个进化问题，它在空间中移动一个表面，使它的面积最快地减小。理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。在数学上这是一个困难的问题，因为这些表面可能会发展奇点，如锥点和撕裂，必须占.

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.