A simple construction of positive loops of Legendrians (Q1721830)

In a manifold \(M\) with a cooriented contact structure \(\xi\), an isotopy of Legendrian submanifolds \(L_t\) is positive if \(\frac{d L_t}{dt}\) points towards the positive side of \(\xi\) for all \(t\) and at all points of \(L_t\). G. \textit{G. Liu} [Positive Legendrian isotopies for loose Legendrian submanifolds. (PhD Thesis); ``On positive loops of loose Legendrian embeddings'', Preprint, \url{arXiv:1605.07494}] has proved that any loose Legendrian \(L\) in the sense of Murphy admits a contractible positive loop of Legendrians. The paper under review gives an alternative proof of this result under some further topological condition on \(L\). The strategy is to use a simple explicit loop of contactomorphisms in \(M\times D^2\) (rotation on the \(D^2\) factor) and then apply the h-principle to find such a submanifold containing \(L\). Some mild topological assumptions on \(L\) are required for the latter step.