Variational Problems in Symplectic and Kahler Geometry

摘要本文将与R. Schoen合作，继续研究拉格朗日子流形的变分问题：设M为具有相容度量的不对称2n流形。给定M中一个可以由浸入式拉格朗日子流形表示的同调类，在表示该同调类的所有拉格朗日量中找到一个体积最小的拉格朗日量。已知最小值以拉格朗日可整流变量的形式存在，因此我们研究的重点是这些最小值的正则性。当环境流形M为Kaehler-Einstein时，就有可能证明拉格朗日极小值在经典意义上是平稳的。我们希望建立一个良好的拉格朗日极小值的正则性理论，然后利用这些极小值来研究环境卡勒流形或辛流形的几何问题。许多物理和数学问题的解都是通过最优化过程得到的。在这个项目中，我们将研究微分几何中的最小化过程。我们在满足几何约束（拉格朗日）的广义曲面（浸入子流形）中最小化体积。因为我们要求广义曲面满足这个约束条件，所以出现了许多新的困难。然而，我们的解通常具有有趣而重要的几何性质。该问题与非线性弹性问题有许多共同的特点。希望我们的技术在这个问题上有用。然而，应该强调的是，在存在几何约束的情况下，优化程序还没有研究过。这个想法最终将在许多不同的环境中发挥作用。

AbstractProposal: DMS 9802487Principal Investigator: Jon WolfsonIn this proposal we will continue our study, joint with R. Schoen, ofvariational problems for lagrangian submanifolds: Let M be asymplectic 2n-manifold equipped with a compatible metric. Given ahomology class in M that can be represented by an immersed lagrangiansubmanifold, find a lagrangian that minimizes volume among alllagrangians that represent this homology class. The minimizer isknown to exist as a lagrangian rectifiable varifold, so the focus ofour research is on the regularity of these minimizers. When theambient manifold M is Kaehler-Einstein it may be possible to show thatthe lagrangian minimizers are stationary, in the classical sense. Wehope to develop a good regularity theory of lagrangian minimizers andthen to use these minimizers to study the geometry of the ambientKaehler or symplectic manifolds.Many problems in physics and mathematics have solutions that are obtained via an optimization procedure. In this project we will study a minimizing procedure in differential geometry. We minimize volume among generalized surfaces (immersed submanifolds) that satisfy a geometric constraint (that are lagrangian). Because we require our generalized surfaces to satisfy this constraint many new difficulties arise. However our solutions often have interesting and important geometric properties. This problem has many features in common with problems in nonlinear elasticity. It is hoped that our techniques will be of use in that subject. However it should be emphasized that optimization procedures in the presence of geometric constraints have not been studied before. This idea will eventually be useful in many different contexts.