Variational Problems in Symplectic and Kahler Geometry

辛几何和卡勒几何中的变分问题

• 批准号: 9802487
• 负责人: Jon Wolfson
• 金额: $ 9.89万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802487&HistoricalAwards=false
• 关键词: Variational; Problems; Symplectic; Kahler; Geometry

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.

摘要提案：DMS 9802487 主要研究员：Jon Wolfson 在本提案中，我们将与 R. Schoen 一起继续研究拉格朗日子流形的变分问题：令 M 为配备兼容度量的渐近 2n 流形。 给定 M 中可以由浸没拉格朗日子流形表示的同调类，找到一个拉格朗日，使表示该同调类的所有拉格朗日量最小化。 众所周知，最小化器以拉格朗日可整流倍数的形式存在，因此我们研究的重点是这些最小化器的正则性。 当环境流形 M 是凯勒-爱因斯坦时，可以证明拉格朗日极小值在经典意义上是平稳的。 我们希望发展出良好的拉格朗日极小值正则理论，然后使用这些极小值来研究环境凯勒或辛流形的几何形状。物理和数学中的许多问题都有通过优化过程获得的解决方案。 在这个项目中，我们将研究微分几何中的最小化过程。 我们最小化满足几何约束（拉格朗日）的广义曲面（浸没子流形）之间的体积。 因为我们需要广义曲面来满足这个约束，所以出现了许多新的困难。 然而，我们的解决方案通常具有有趣且重要的几何特性。 该问题与非线性弹性问题有许多共同点。 希望我们的技术能在该主题中有用。 然而，应该强调的是，之前尚未研究过存在几何约束的优化过程。 这个想法最终将在许多不同的环境中发挥作用。