On Legendrian embeddings into open book decompositions

We study Legendrian embeddings of a compact Legendrian submanifold $L$ sitting in a closed contact manifold $(M,\xi)$ whose contact structure is supported by a (contact) open book $\mathcal{OB}$ on $M$. We prove that if $\mathcal{OB}$ has Weinstein pages, then there exist a contact structure $\xi^{\prime}$ on $M$, isotopic to $\xi$ and supported by $\mathcal{OB}$, and a contactomorphism $f : (M, \xi) \to (M, \xi^{\prime})$ such that the image $f(L)$ of any such submanifold can be Legendrian isotoped so that it becomes disjoint from the closure of a page of $\mathcal{OB}$.

Keywords

contact, convex symplectic, Weinstein, Liouville, Lefschetz fibration, open book

2010 Mathematics Subject Classification

57R65, 58A05, 58D27

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The first author is partially supported by NSF FRG grant DMS-0905917. The second author is partially supported by NSF FRG grant DMS-1065910, and also by TUBITAK grant 1109B321200181.

Received 2 April 2018

Received revised 2 February 2019

Accepted 19 February 2019

Published 7 October 2019