Positive isotopies of Legendrian submanifolds and applications

TL;DR

This paper proves the non-existence of positive loops in certain Legendrian embedding spaces under specific conditions and explores related results and applications in contact topology.

Contribution

It establishes new non-existence results for positive loops of Legendrian submanifolds in particular contact manifolds and applies these findings to isotopies near surfaces in tight contact 3-manifolds.

Findings

No positive loop inside the fiber component in Legendrian embedding space when the universal cover of M is Euclidean.

Results on positive isotopies in the space of one-jets of functions.

Application to positive isotopies near surfaces in tight contact 3-manifolds.

Abstract

We show that there is no positive loop inside the component of a fiber in the space of Legendrian embeddings in the contact manifold $ST^*M$, provided that the universal cover of $M$ is $\RM^n$. We consider some related results in the space of one-jets of functions on a compact manifold. We give an application to the positive isotopies in homogeneous neighborhoods of surfaces in a tight contact 3-manifold.