Variational Problems in Symplectic and Kahler Geometry

摘要建议：DMS 9802487主要研究者：乔恩·沃尔夫森在这个建议中，我们将继续我们的研究，与R. Schoen关于Lagrange子流形变分问题的研究：设M是具有相容度量的2n-辛流形。 给定M中的一个同调类，它可以由一个浸入的拉格朗日子流形表示，在表示这个同调类的所有拉格朗日量中，找到一个使体积最小的拉格朗日量。 由于极小元是以拉格朗日可求长变分形式存在的，所以我们的研究重点是这些极小元的正则性。 当周围的流形M是Kaehler-Einstein的时候，在经典意义上，我们可以证明拉格朗日极小元是平稳的。 我们希望发展一个良好的正则性理论的拉格朗日极小，然后使用这些极小来研究几何的周围Kaehler或辛流形，在物理和数学中的许多问题的解决方案，通过一个优化过程。 在这个项目中，我们将研究微分几何中的极小化过程。 我们最小化广义曲面（浸入子流形），满足几何约束（拉格朗日）之间的体积。 因为我们要求我们的广义曲面满足这个约束，许多新的困难出现。 然而，我们的解决方案往往有有趣的和重要的几何性质。 这个问题与非线性弹性力学问题有许多共同的特点。 希望我们的技术能在这方面有所帮助。 然而，应该强调的是，在几何约束的存在下的优化程序还没有被研究过。 这个想法最终将在许多不同的背景下有用。

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.