Topology of Legendrian and minimal submanifolds

University Of Southern California, Los Angeles CA

Investigators

Abstract

The project concerns contact geometry and the geometry of minimal varieties. The soft properties of contact geometry are governed by so called h-principles. In recent years, the hard properties have been uncovered by using holomorphic curve techniques in the framework of Symplectic Field Theory. The project proposes to study a part of this theory known as Legendrian contact homology. This theory has had enormous success for Legendrian knots of dimension 1. Parallels of the 1-dimensional phenomena have been shown to exists for higher dimensional Legendrian submanifolds but the effectiveness of contact homology in higher dimensions has been limited because computations in the theory are comparatively difficult since they involve infinite dimensional spaces in an essential way. One of the main goals of this research project is to prove a conjecture which reduces the computation of Legendrian contact homology in 1-jet spaces to a purely finite dimensional problem. This would be important not only for contact geometry itself: profound recent results in knot theory were derived using heuristic arguments from higher dimensional contact homology and a proof of the conjecture would establish the link between knot theory and higher dimensional contact homology rigorously. The conjecture will also be applied both to internal questions in contact geometry (e.g. to estimate the number of double points of exact Lagrangian immersions in $\C^n$) and to problems in differential topology. The project also intends to complete earlier results concerning minimal surfaces with small total boundary curvature as well as expand the range of applications of the techniques used there, in particular, to problems concerning higher dimensional minimal varieties. Many of the problems arising in connection with this study asks for topological constructions with geometrical control. In topology one is often concerned with open differential relations and the class of allowed deformations is very large. This is a reflection of the fact that spaces studied in topology in a sense are "soft" objects. In geometry, on the other hand, one often faces differential equations and the class of deformations is considerably smaller. Comparing to the situation in topology, one could say that objects in geometry are "hard". This project proposes to study problems in the two areas, contact geometry and minimal varieties, using the interplay between soft and hard.