Topology of Legendrian and minimal submanifolds

Legendrian 拓扑和最小子流形

• 批准号: 0505076
• 负责人: Ko Honda
• 金额: --
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505076&HistoricalAwards=false
• 关键词: Topology; Legendrian; minimal; submanifolds

The project concerns contact geometry and the geometry of minimal varieties.The soft properties of contact geometry are governed by so calledh-principles. In recent years, the hard properties have been uncovered byusing holomorphic curve techniques in the framework of Symplectic FieldTheory. The project proposes to study a part of this theory known asLegendrian contact homology. This theory has had enormous success forLegendrian knots of dimension 1. Parallels of the 1-dimensional phenomenahave been shown to exists for higher dimensional Legendrian submanifoldsbut the effectiveness of contact homology in higher dimensions has beenlimited because computations in the theory are comparatively difficultsince they involve infinite dimensional spaces in an essential way. One ofthe main goals of this research project is to prove a conjecture whichreduces the computation of Legendrian contact homology in 1-jet spaces toa purely finite dimensional problem. This would be important not only forcontact geometry itself: profound recent results in knot theory werederived using heuristic arguments from higher dimensional contact homologyand a proof of the conjecture would establish the link between knot theoryand higher dimensional contact homology rigorously. The conjecture willalso be applied both to internal questions in contact geometry (e.g. toestimate the number of double points of exact Lagrangian immersions in$\C^n$) and to problems in differential topology. The project also intendsto complete earlier results concerning minimal surfaces with small totalboundary curvature as well as expand the range of applications of thetechniques used there, in particular, to problems concerning higherdimensional minimal varieties. Many of the problems arising in connectionwith this study asks for topological constructions with geometricalcontrol.In topology one is often concerned with open differential relations andthe class of allowed deformations is very large. This is a reflection ofthe fact that spaces studied in topology in a sense are "soft" objects. Ingeometry, on the other hand, one often faces differential equations andthe class of deformations is considerably smaller. Comparing to thesituation in topology, one could say that objects in geometry are "hard".This project proposes to study problems in the two areas, contact geometryand minimal varieties, using the interplay between soft and hard.

该项目涉及接触几何和最小簇几何。接触几何的软属性由所谓的h-原理支配。近年来，在辛场论的框架下，利用全纯曲线技巧揭示了硬性质。该项目建议研究这一理论的一部分被称为Legendrian接触同源性。这个理论在一维的勒让德结上取得了巨大的成功。一维现象的相似性已经被证明存在于高维的Legendrian子流形中，但是接触同调在高维中的有效性受到了限制，因为理论中的计算相对困难，因为它们基本上涉及无限维空间。本研究项目的主要目标之一是证明一个猜想，该猜想将1-jet空间中Legendrian接触同调的计算简化为一个纯粹的有限维问题。这不仅对接触几何本身是重要的：纽结理论中的深刻的最新结果是使用来自高维接触同调的启发式论证导出的，并且猜想的证明将严格地建立纽结理论和高维接触同调之间的联系。该猜想也将被应用于接触几何的内部问题（例如，估计$\C^n$中精确拉格朗日浸入的双重点的数量）和微分拓扑学的问题。该项目还打算完成较早的结果，关于极小曲面与小的总边界曲率，以及扩大的应用范围，使用的技术，特别是有关高维极小品种的问题。许多与本研究有关的问题都需要几何控制的拓扑结构，在拓扑学中，人们常常关心的是开微分关系，允许的变形类是非常大的.这反映了拓扑学中研究的空间在某种意义上是“软”对象的事实。另一方面，在几何学中，人们经常面对微分方程，而变形的种类要小得多。与拓扑学中的情况相比，可以说几何学中的对象是“硬”的。本项目提出利用软硬之间的相互作用来研究接触几何和最小变化这两个领域的问题。