Topics in Lagrangian Geometry

拉格朗日几何专题

• 批准号: 0304587
• 负责人: Jon Wolfson
• 金额: $ 17.5万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0304587&HistoricalAwards=false
• 关键词: Topics; Lagrangian; Geometry

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.

项目摘要标题：主题在拉格朗日几何该项目将调查的存在问题的特殊拉格朗日子流形在卡-丘流形使用变分技术和使用平均曲率流。变分技术涉及拉格朗日循环的约束变分问题的研究。这些问题可以制定在任意Kaehler（或辛）流形，并涉及最小化体积之间的拉格朗日，代表一个固定的同源类和证明最佳正则性的极小。特别是该项目希望表明，如果周围的流形是一个卡-丘3 $倍，然后，对于一个适当制定的问题，极小是特殊的拉格朗日。长期以来，人们一直知道，如果环境流形是凯勒-爱因斯坦流形，则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，它打算调查的条件下，流动和不发展奇点在有限的time. Variation问题的几何约束和平均曲率流的余维大于一个是数学分析的前沿。这些问题在几何中是自然的，但在许多不同的应用问题中也很重要。在材料科学中，“模型”问题要求在圆盘之间的面积保持映射中找到“动能”的最小值，并找到最小值的最佳平滑度。目前，极小元的存在性是已知的，但对它的奇点一无所知。这个项目的部分内容与这种“规则”问题密切相关。在弦理论中，著名的工作证明了一类体积极小的三维曲面的存在，称为特殊拉格朗日子流形。这个项目是一个直接的尝试，以肯定的方式回答这个问题。 不同余维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”约束来理解更高余维中的平均曲率流。这个主题相对来说还没有被探索过。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。