Geometric Variational Problems

摘要-DMS-0104007-几何变分问题主要研究人员建议与R.舍恩共同继续研究拉格朗日环的约束变分问题。在其最基本的形式下，问题可以被提出如下：考虑具有与辛形式相容的度量的辛流形。修正了一个可以用拉格朗日圈表示的同调类。在代表这类问题的所有拉格朗日循环中找出一个体积最小的拉格朗日循环，并推导出这个循环的最优正则性。如果辛流形是Kaehler，并且具有Kaehler-Einstein度量，则极小元的充分正则性意味着极小元既是拉格朗日的，又是极小的(零中曲率)。如果第一个Chern类是负的，在适当的意义上，这样的子流形可能是唯一的，然后对于理解环境流形的几何是有用的。如果Kaehler流形是一个Calabi-Yau流形，充分正则性蕴含着极小子流形是一个有标子流形，一个特殊的Lagrangian子流形.我们建议研究这个变分问题和相关的变分问题的存在性和正则性，并研究这些结果对Kaehler-Einstein流形几何的影响.这个结果得到了一个Calabi-Yau流形的特殊Lagrange子流形的存在定理.这个结果是Strominger-Yau-Zaslow为几何构造“镜像对称”而提出的程序的一个重要部分.镜像对称性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。其核心提出了一类称为Calabi-Yau流形的流形之间的对偶。这种二元性允许在一个流形上执行计算，从而产生其“镜像”的结果。镜面对称性的实现将影响到代数几何、微分几何、拓扑学、偏微分方程式和弦理论等多门学科。目前对这门学科的兴趣有助于弥合物理和数学之间的鸿沟。在二维中，我们的建议与著名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，这是第一次对几何约束变分问题进行系统研究的尝试。这个想法还将在几何学及其应用中有其他重要的应用。

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.