Variational Problems in Symplectic and Kahler Geometry

摘要建议：DMS 9802487主要研究者：Jon Wolfson在这项建议中，我们将继续与R.Schoen一起研究拉格朗日子流形的变分问题：设M是配备相容度量的不对称2n流形。给定M中可由浸没的拉格朗日子流形表示的非同调类，在表示该同调类的所有拉格朗日中找到一个最小化体积的拉格朗日。由于极小元是以拉格朗日可导变量的形式存在的，所以我们研究的重点是这些极小元的正则性。当环境流形M是Kaehler-Einstein时，就有可能证明拉格朗日极小化子在经典意义上是稳定的。我们希望发展一个很好的拉格朗日极小值的正则性理论，然后用这些极小值来研究环境Kaehler或辛流形的几何。物理和数学中的许多问题都有通过优化过程得到的解。在这个项目中，我们将研究微分几何中的一个极小化过程。我们最小化满足几何约束(拉格朗日约束)的广义曲面(浸入子流形)之间的体积。因为我们要求我们的广义曲面满足这个约束，所以出现了许多新的困难。然而，我们的解通常具有有趣且重要的几何性质。这一问题与非线性弹性力学问题有许多共同之处。希望我们的技术将在这个问题上有所用处。然而，应该强调的是，在存在几何约束的情况下，优化过程以前从未被研究过。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.