Topics in Lagrangian Geometry

项目摘要标题：拉格朗日几何中的主题该项目将使用变分技术和平均曲率流来研究Calabi-Yau流形中特殊的拉格朗日子流形的存在性问题。变分技术涉及到拉格朗日环的约束变分问题的研究。这些问题可以表示在任意的Kaehler(或辛流形)中，并且涉及表示固定同调类的拉格朗日之间的体积最小化以及证明极小化的最优正则性。特别是，该项目希望证明，如果环境流形是一个Calabi-Yau$3$倍数，那么对于一个适当的公式，极小元是特殊的拉格朗日函数。众所周知，如果周围流形是Kaehler-Einstein，则拉格朗日子流形的平均曲率流保持拉格朗日条件。本项目旨在研究Kaehler-Einstein流形的拉格朗日子流形的平均曲率流的正则性。特别是，它打算研究流在有限时间内发生奇点和不发生奇点的条件。具有几何约束和余维平均曲率流大于1的变分问题是数学分析的前沿。这些问题在几何学中是很自然的，但在许多不同的应用问题中也很重要。在材料科学中，一个“模型”问题要求在圆盘之间的面积保持映射中找到一个“动能”的最小化，并找到该最小化的最佳光滑度。目前，极小值的存在是已知的，但对它的奇点却一无所知。这项工程的部分内容与这种“规律性”问题密切相关。在弦论中，著名的工作猜想存在一类体积最小的三维曲面，称为特殊的拉格朗日子流形。这个项目是对这个问题的正面回答的直接尝试。各种余维中的平均曲率流模拟了许多不同的物理现象。这个项目试图利用“拉格朗日”约束来理解高余维中的平均曲率流。这一主题相对来说还没有被探索过。这个项目中研究的技术有望加强几何与应用数学和工程的各个领域之间的互动，并将新的结果和技术引入这些领域。

Project AbstractTitle: TOPICS IN LAGRANGIAN GEOMETRYThe project will investigate the question of the existence of special lagrangian submanifolds in Calabi-Yau manifolds using variational techniques and using mean curvature flow. The variational techniques involve the study of constrained variational problems for lagrangian cycles. These problems can be formulated in arbitrary Kaehler (or symplectic) manifolds and involve minimizing volume among lagrangians that represent a fixed homology class and proving optimal regularity of the minimizer. In particular the project hopes to show that if the ambient manifold is a Calabi-Yau $3$-fold then, for a suitably formulated problem, the minimizer is special lagrangian. It has long been known that the mean curvature flow of a lagrangian submanifold preserves the lagrangian condition if the ambient manifold is Kaehler-Einstein. The project intends to investigate the regularity properties of the mean curvature flow of a lagrangian submanifold of a Kaehler-Einstein manifold. In particular, it intends to investigate the conditions under which the flow does and does not develop singularities in finite time.Variational problems with geometric constraints and mean curvature flow in codimension greater than one are on the frontier of mathematical analysis. These problems are natural in geometry but they are also important in many different applied problems. In material science a ``model'' problem asks to find a minimizer of ``kinetic energy'' among area preserving maps between disks and to find the optimal smoothness of the minimizer. At present, the existence of a minimizer is known but nothing is known about its singularities. Parts of this project are closely related to this kind of ``regularity'' question. In string theory well known work conjectures the existence of a certain class of volume minimizing three dimensional surfaces called special lagrangian submanifolds. This project is a direct attempt to answer this question in the affirmative. Mean curvature flow in various codimensions models many different physical phenomena. This project attempts to exploit the ``lagrangian'' constraint to get an understanding of mean curvature flow in higher codimensions. This subject is relatively unexplored. The techniques investigated in this project hold the promise of enhancing the interaction between geometry and various fields of applied mathematics and engineering and in bringing new results and techniques into these fields.