Topology of Legendrian and minimal submanifolds

点击翻译按钮获取中文摘要

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.