Topics in Differential Geometry

微分几何专题

• 批准号: 0604759
• 负责人: Jon Wolfson
• 金额: $ 19.7万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-15 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604759&HistoricalAwards=false
• 关键词: Topics; Differential; Geometry

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.

AbstractAward：DMS 0604759首席研究员：Jon Wolfson该项目将使用变分和变形技术并使用平均曲率流来研究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。"拉格朗日霍奇猜想“指出，给定卡-丘流形中的一个整数拉格朗日同调类，存在一个特殊的拉格朗日环来代表这个类。这个猜想在Calabi-Yau曲面（K3曲面）中是错误的。的失败是密切相关的奇异性的存在（而不是分支点）最小化拉格朗日。 本项目的目的是回答一个相关的猜想，“变形拉格朗日霍奇猜想”：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，并给定一个拉格朗日同调类，假设族中的初始度量允许一个特殊的拉格朗日环，该拉格朗日环表示拉格朗日同调类。然后，我们猜想，在theone-parameter家庭的度量足够接近的初始度量也承认specialagrangian循环，代表这一类。这个猜想在二维情形（K3曲面中的特殊拉格朗日曲面的情形）和初始特殊拉格朗日量为无穷大的情形下成立。已知光滑拉格朗日子流形的平均曲率流保持拉格朗日条件，如果环境流形是Kaehler-Einstein。该项目的目的是研究拉格朗日函数在这种流动下的行为的各种问题，特别是，研究可以用来给出反例的例子，以证明任何这样的反例必须是全局的。具有几何约束的变分问题，变形问题和余维大于1的平均曲率流动是数学分析的前沿。这些问题在几何学中是自然的，但在许多不同的应用问题中也是重要的.例如，在材料科学中，一个模型问题要求找到一个最小化的“动能”保留磁盘之间的映射区域，并找到最小化的最佳平滑度。 本项目的部分内容与这种存在性和规律性问题密切相关，因此与发展非线性弹性理论基础上的技术密切相关。在理论物理学的分支弦理论中，著名的工作证明了存在一类体积极小的三维曲面，称为特殊拉格朗日子流形。这个项目是解决这个猜想的一个直接尝试。 不同余维的平均曲率流模拟了许多不同的物理现象，包括火焰传播。这个项目试图利用"拉格朗日“约束来理解更高余维中的平均曲率流。在这个项目中调查的技术举行的承诺，加强几何和应用数学和工程的各个领域之间的相互作用，并在这些领域带来新的成果和技术。