Topics in Differential Geometry

摘要奖：DMS 0604759首席研究员：Jon Wolfson该项目将使用变分和变形技术以及平均曲率流研究Kaehler-Einstein流形中极小拉格朗日子流形和Calabi-Yau流形中特殊拉格朗日子流形的存在性问题。拉格朗日-霍奇猜想指出，给定Calabi-Yau流形中的一个积分拉格朗日同调类，存在一个表示这一类的特殊拉格朗日圈。这一猜想现在在Calabi-Yau曲面(K3曲面)上是错误的。猜想的失败与最小化拉格朗日的奇点(而不是分支点)的存在密切相关。这个项目的目的是回答一个相关的猜想，“形变拉格朗日Hodge猜想”：给定一个单参数的Calabi-Yau度量族，其相关的Kaehler形式具有固定的上同调类，并给出一个拉格朗日同调类，假设这个族中的初始度量允许一个特殊的拉格朗日圈代表拉格朗日同调类。然后，我们猜想，单参数族中足够接近初始度量的度量也允许代表这一类的特殊的拉格朗日循环。这一猜想已在二维情形(K3曲面中特殊拉格朗日曲面的情形)和初始特殊拉格朗日为动画的情况下成立。众所周知，如果环境流形是Kaehler-Einstein，则光滑拉格朗日子流形的平均曲率流保持拉格朗日条件。该项目旨在研究在这种流动下拉格朗日人的行为的各种问题，特别是研究可以用来对Thomas-Yau猜想给出反例或证明任何这样的反例一定是全球性的例子。具有余维大于1的几何约束、变形问题和平均曲率流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。例如，在材料科学中，一个模型问题要求在磁盘之间的面积保持映射中找到一个“动能”的最小化，并找到最小化的最佳光滑度。本项目的部分内容与这种存在性和规律性问题密切相关，从而与以非线性弹性理论为基础的发展技术密切相关。在弦理论中，理论物理的分支，众所周知的工作是猜想存在一类体积最小的三维曲面，称为特殊拉格朗日子流形。这个项目是对这一猜想的直接尝试。不同余维中的平均曲率流模拟了许多不同的物理现象，例如，包括火焰传播。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，并将新的成果和技术引入这些领域。

AbstractAward: DMS 0604759Principal Investigator: Jon WolfsonThe project will investigate the question of the existence minimal lagrangiansubmanifolds in Kaehler-Einstein manifolds and special lagrangian submanifoldsin Calabi-Yau manifolds using variational and deformation techniques and usingmean curvature flow. The ``Lagrangian Hodge Conjecture'' states that given anintegral lagrangian homology class in a Calabi-Yau manifold there is a speciallagrangian cycle that represents this class. This conjecture is now known tobe false in a Calabi-Yau surface (a K3 surface). The failure of the conjectureis closely related to the existence of singularities (other than branchpoints) on minimizing lagrangians. This project aims to answer a relatedconjecture, the ``Deformation Lagrangian Hodge Conjecture'': Given aone-parameter family of Calabi-Yau metrics whose associated Kaehler forms havefixed cohomology class and given a lagrangian homology class, suppose that theinitial metric in the family admits a special lagrangian cycle that representsthe lagrangian homology class. Then, we conjecture, that the metrics in theone-parameter family sufficient close to the initial metric also admit speciallagrangian cycles that represents this class. This conjecture has beenestablished in the two dimension case (the case of special lagrangian surfacesin K3 surfaces) and in the case that the initial special lagrangian is animmersion. It is known that the mean curvature flow of a smooth lagrangiansubmanifold preserves the lagrangian condition if the ambient manifold isKaehler-Einstein. The project intends to investigate various questions aboutthe behavior of lagrangians under this flow, in particular, to study examplesthat can be used either to give counter-examples to the Thomas-Yau conjectureor show that any such counter-example must be global.Variational problems with geometric constraints, deformation problems and meancurvature flow in codimension greater than one are on the frontier ofmathematical analysis. These problems are natural in geometry but they arealso important in many different applied problems. For example, in materialscience a model problem asks to find a minimizer of ``kinetic energy'' amongarea preserving maps between disks and to find the optimal smoothness of theminimizer. Parts of this project are closely related to this kind ofexistence and regularity question and therefore to developing techniques atthe foundations of the theory of non-linear elasticity. In string theory, abranch of theoretical physics, well known work conjectures the existence of acertain class of volume minimizing three dimensional surfaces called speciallagrangian submanifolds. This project is a direct attempt to resolve thisconjecture. Mean curvature flow in various codimensions models many differentphysical phenomena, including, for example, flame propagation. This projectattempts to exploit the ``lagrangian'' constraint to get an understanding ofmean curvature flow in higher codimensions. The techniques investigated inthis project hold the promise of enhancing the interaction between geometryand various fields of applied mathematics and engineering and in bringing newresults and techniques into these fields.