Geometric Variational Problems

几何变分问题

• 批准号: 0104007
• 负责人: Jon Wolfson
• 金额: $ 6.68万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104007&HistoricalAwards=false
• 关键词: Geometric; Variational; Problems

Abstract- DMS-0104007-Geometric Variational ProblemsThe principal investigator proposes to continue the study, joint with R. Schoen, of constrained variationalproblems for lagrangian cycles. In its most basic form the problem can be posed as follows: Consider a symplecticmanifold with a metric compatible with the symplectic form. Fix a homology class that can be represented by a lagrangian cycle. Find a lagrangian cycle that minimizes volume among all lagrangian cycles representing this class and derive optimal regularity of this cycle. In the case that the symplectic manifold is Kaehler,with Kaehler-Einstein metric, sufficient regularity of the minimizer implies that the minimizer is both lagrangian and minimal (zero mean curvature). If the first Chern class is negative such submanifolds could be unique, in a suitable sense,and then useful in understanding the geometry of the ambient manifold. If the Kaehler manifold is a Calabi-Yau manifold sufficient regularity implies that the minimizer is a calibrated submanifold, a special lagrangian submanifold.We propose to investigate the existence and regularity of this and related varitional problems and to study the consequences of these results on thegeometry of Kaehler-Einstein manifolds.A consequence of the proposal is an existence theorem for special lagrangian submanifolds of a Calabi-Yau manifold.This result is an essential part of the program proposed by Strominger-Yau-Zaslow for the geometric construction of "mirrorsymmetry". Mirror symmetry is one of the most interesting and important problems currently being studied in mathematics and theoretical physics. At its core it proposes a duality between a class of manifolds called Calabi-Yau manifolds. This duality allows computations to be performed on one manifold that yield a result for its "mirror". Thus computations that are otherwise extremely difficult can be achieved.The realization of mirror symmetry will effect such diverse subjects as algebraic geometry, differential geometry, topology, partial differential equations and string theory.The current interest in this subject helps bridge the gap between physics and mathematics. In two-dimensions our proposal has some close analogies to a well-known model problemin non-linear elasticity. The regularity theory developed here will shed light on the difficult regularity problems of that theory. Finally this problem is the first attempt to make a systematic study of a variational problem with a geometric constaint. This idea will have other important applications in geometry and its applications.

摘要- DMS-0104007-几何变分问题首席研究员建议与 R. Schoen 一起继续研究拉格朗日循环的约束变分问题。就其最基本的形式而言，该问题可以提出如下：考虑具有与辛形式兼容的度量的辛流形。修复可以用拉格朗日循环表示的同源类。找到一个拉格朗日循环，使代表此类的所有拉格朗日循环中的体积最小化，并导出该循环的最佳正则性。在辛流形为凯勒的情况下，采用凯勒-爱因斯坦度量，极小值的充分正则性意味着极小值既是拉格朗日又是极小值（零平均曲率）。如果第一个 Chern 类为负，则此类子流形在适当的意义上可能是唯一的，然后有助于理解环境流形的几何形状。如果凯勒流形是卡拉比-丘流形，则充分规律性意味着极小值是一个校准子流形，即特殊拉格朗日子流形。我们建议研究该问题及相关变分问题的存在性和规律性，并研究这些结果对凯勒-爱因斯坦流形几何的影响。该提议的一个结果是特殊拉格朗日的存在定理 Calabi-Yau 流形的子流形。这个结果是 Strominger-Yau-Zaslow 提出的“镜像对称”几何构造程序的重要组成部分。镜像对称是目前数学和理论物理研究中最有趣和最重要的问题之一。它的核心提出了称为 Calabi-Yau 流形的一类流形之间的对偶性。这种对偶性允许在一个流形上执行计算，产生其“镜像”的结果。从而可以实现原本极其困难的计算。镜像对称的实现将影响代数几何、微分几何、拓扑、偏微分方程和弦理论等不同学科。当前对这一学科的兴趣有助于弥合物理和数学之间的差距。在二维中，我们的建议与众所周知的非线性弹性模型问题有一些密切的相似之处。这里发展的正则性理论将阐明该理论中困难的正则性问题。最后，该问题是对具有几何约束的变分问题进行系统研究的首次尝试。这个想法将在几何及其应用中具有其他重要的应用。