Topics in Differential Geometry

项目主要研究Kaehler-Einstein流形中的最小拉格朗日子流形和Calabi-Yau流形中的特殊拉格朗日子流形的存在性问题，采用变分和变形技术，利用平均曲率流。拉格朗日霍奇猜想（Lagrangian Hodge Conjecture）指出，给定Calabi-Yau流形中的一个积分拉格朗日同调类，存在一个特殊的拉格朗日循环来表示这个类。这个猜想现在已知在Calabi-Yau曲面（K3曲面）中是错误的。该猜想的失败与最小化拉格朗日的奇异性（分支点除外）的存在密切相关。本项目旨在回答一个相关的猜想，“变形拉格朗日霍奇猜想”：给定一个卡拉比-丘度量的单参数族，其相关的Kaehler形式具有固定的上同调类，并给定一个拉格朗日同调类，假设该族中的初始度量允许一个特殊的拉格朗日循环，该拉格朗日同调类。然后，我们推测，在足够接近初始度规的单参数族中的度规也承认表示这类的特殊的拉格朗日环。这个猜想已经在二维情况下（K3曲面中的特殊拉格朗日曲面）和初始特殊拉格朗日为浸入的情况下得到了建立。已知当周围流形为kaehler - einstein流形时，光滑拉格朗日子流形的平均曲率流保持拉格朗日条件。该项目打算调查关于拉格朗日量在这种流下的行为的各种问题，特别是研究可以用来给出托马斯-丘猜想的反例或表明任何这种反例必须是全局的例子。具有几何约束的变分问题、变形问题和余维大于1的曲率流是数学分析的前沿问题。这些问题在几何中是很自然的，但它们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区域保持映射中找到“动能”的最小值，并找到最小值的最佳平滑度。本项目的部分内容与这种存在性和规律性问题密切相关，因此与在非线性弹性理论基础上开发技术密切相关。在理论物理的一个分支——弦理论中，著名的工作推测存在一类体积最小化的三维曲面，称为特拉格朗日子流形。这个项目是解决这个猜想的直接尝试。平均曲率流在各种共维模型许多不同的物理现象，包括，例如，火焰传播。这个项目试图利用“拉格朗日”约束来理解高协维的平均曲率流。本项目所研究的技术有望加强几何与应用数学和工程各个领域之间的相互作用，并为这些领域带来新的成果和技术。

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.