Topology of Legendrian and minimal submanifolds

该项目涉及接触几何和最小方差几何。接触几何的软性质受所谓的h-原理的支配。近年来，在辛场理论的框架下，利用全纯曲线技术揭示了材料的硬性质。该项目建议研究这一理论的一部分，称为勒让德接触同调。这一理论对一维Legendrian子流形已经取得了巨大的成功。高维Legendrian子流形也存在与一维现象类似的现象，但高维接触同调的有效性是有限的，因为理论中的计算相对困难，因为它们本质上涉及无限维空间。这个研究项目的主要目的之一是证明一个猜想，该猜想将1-JET空间中的Legendrian联络同调的计算简化为一个纯有限维问题。这不仅对接触几何本身很重要：纽结理论中深刻的最新结果是利用高维接触同调的启发式论证而得到的，该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。这个猜想也将应用于接触几何的内部问题(例如估计$C^n中的精确拉格朗日浸没的双点个数)和微分拓扑学中的问题。该项目还旨在完善关于总边界曲率较小的极小曲面的早期结果，并扩大那里使用的技术的应用范围，特别是关于高维极小变差的问题。在这项研究中出现的许多问题都要求具有几何控制的拓扑结构。在拓扑学中，人们经常涉及开放的微分关系，并且允许的变形的类别非常大。这反映了拓扑学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，人们经常要面对的是微分方程式，变形的种类要小得多。与拓扑学的现状相比，几何中的对象可以说是“硬的”。本项目提出利用软和硬的相互作用来研究接触几何和最小变差这两个领域的问题。

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.