Geometric Variational Problems

摘要- dms -0104007-几何变分问题主要研究者提议与R. Schoen一起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，问题可以提出如下：考虑一个辛流形，其度量与辛形式相容。修正一个可以用拉格朗日循环表示的同调类。在所有代表这一类的拉格朗日循环中找到一个体积最小的拉格朗日循环，并推导出该循环的最优正则性。当辛流形为Kaehler，且具有Kaehler- einstein度规时，极小量的充分正则性意味着极小量既是拉格朗日的，又是极小的（平均曲率为零）。如果第一个Chern类是负的，这样的子流形在适当的意义上可能是唯一的，然后对理解周围流形的几何有用。如果Kaehler流形是Calabi-Yau流形，则充分正则性意味着最小化器是一个校准子流形，一个特殊的拉格朗日子流形。我们打算研究这个和相关的变分问题的存在性和规律性，并研究这些结果对Kaehler-Einstein流形几何的影响。该建议的一个结果是Calabi-Yau流形的特殊拉格朗日子流形的存在性定理。这一结果是strominger - yu - zaslow提出的“镜像对称”几何构造方案的重要组成部分。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。其核心是提出了一类叫做Calabi-Yau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，从而为其“镜像”产生结果。这样就可以实现原本极其困难的计算。镜像对称的实现将影响到代数几何、微分几何、拓扑学、偏微分方程和弦理论等多种学科。当前对这门学科的兴趣有助于弥合物理学和数学之间的差距。在二维中，我们的建议与一个众所周知的非线性弹性模型问题有一些相似之处。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，这个问题是对几何约束的变分问题进行系统研究的第一次尝试。这个思想在几何学及其应用中还有其他重要的应用。

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.