Topology of Legendrian and minimal submanifolds

该项目涉及接触几何和最小变化几何。接触几何的软特性是由所谓的原理支配的。近年来，在辛场论的框架下，利用全纯曲线技术揭示了硬性质。该项目建议研究这一理论的一部分，称为legendrian接触同源。这个理论对于1维的legendrian结已经取得了巨大的成功。一维现象的平行线已被证明存在于高维的Legendrian子流形中，但接触同调在高维中的有效性受到限制，因为理论中的计算相对困难，因为它们在本质上涉及无限维空间。本研究项目的主要目标之一是证明一个猜想，该猜想将1-射流空间中的Legendrian接触同调的计算简化为纯粹的有限维问题。这不仅对接触几何本身很重要：最近结理论的深刻结果是使用高维接触同调的启发式论证推导出来的，并且该猜想的证明将严格地建立结理论和高维接触同调之间的联系。这个猜想也将被应用于接触几何中的内部问题（例如，估计精确拉格朗日浸没在C^n$中的双点的数量）和微分拓扑问题。该项目还打算完成关于具有小总边界曲率的最小曲面的早期结果，并扩大在那里使用的技术的应用范围，特别是关于高维最小变量的问题。与本研究相关的许多问题都要求具有几何控制的拓扑结构。在拓扑学中，人们经常关注开微分关系，允许变形的种类非常大。这反映了一个事实，即拓扑研究的空间在某种意义上是“软”对象。另一方面，几何学经常面对微分方程，而且变形的种类要小得多。与拓扑学中的情况相比，人们可以说几何中的对象是“硬”的。本项目拟利用软硬相互作用研究接触几何和最小变异两个方面的问题。

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.