Non-simplicity of Isocontact Embeddings in All Higher Dimensions

Abstract

In this article we show that in any dimension there exist infinitely many pairs of formally contact isotopic isocontact embeddings into the standard contact sphere which are not contact isotopic. This is the first example of rigidity for contact submanifolds in higher dimensions. The contact embeddings are constructed via contact push-offs of higherdimensional Legendrian submanifolds.

Infinite not contact isotopic embeddings in $(S^{2n-1},ξ_{\mathrm{std}})$ for $n\ge 4$

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For n ≥ 4, we show that there are infinitely many formally contact isotopic embeddings of the standard ST * S n−1 to (S 2n−1 , ξ std ) that are not contact isotopic. This answers a conjecture of Casals and Etnyre [5] except for the n = 3 case. The argument does not appeal to the surgery formulae of critical handle attachement for Floer theory/SFT.

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“…Their higher dimensional analogues are less studied until recently. A recent breakthrough in this direction is due to Casals and Etnyre [5], where they proved that isocontact embeddings of codimension 2 submanifolds is not simple, i.e. being contact isotopic is finer than the underlying topological information (being formally contact isotopic).…”

“…Such rigidity result shows the fundamental difference with isocontact embeddings of codimension at least 4 submanifolds, which are governed by h-principle [9], hence completely determined by their formal data. More precisely, Casals and Etnyre [5,Theorem 1.1] showed that there are two contact embeddings of the standard ST * S n−1 into (S 2n−1 , ξ std ) for n ≥ 3 that are formally contact isotopic but not contact isotopic. Then they conjectured [5,Conjecture 1.5] that there are infinitely many formally contact isotopic embeddings of the standard ST * S n−1 into (S 2n−1 , ξ std ) that are not contact isotopic to each other.…”

“…More precisely, Casals and Etnyre [5,Theorem 1.1] showed that there are two contact embeddings of the standard ST * S n−1 into (S 2n−1 , ξ std ) for n ≥ 3 that are formally contact isotopic but not contact isotopic. Then they conjectured [5,Conjecture 1.5] that there are infinitely many formally contact isotopic embeddings of the standard ST * S n−1 into (S 2n−1 , ξ std ) that are not contact isotopic to each other. In this note, we give a proof of their conjecture for n ≥ 4.…”

“…Let us briefly recall the geometric constructions in [5]. Starting from a Legendrian sphere Λ in (S 2n−1 , ξ std ), using the Weinstein neighborhood theorem, we get a contact pushoff, i.e.…”