Some non-collarable slices of Lagrangian surfaces
type: Item
instance of: scholarly article
title: Some non-collarable slices of Lagrangian surfaces
zbMATH Open document ID: 1251.57017
author: Baptiste Chantraine
published in: Bulletin of the London Mathematical Society
publication date: 2012-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.2969
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 53D12
zbMATH DE Number: 6092629
zbMATH Keywords: symplectizations
zbMATH Keywords: contact hypersurfaces
zbMATH Keywords: Lagrangian submanifolds
MaRDI profile type: Publication
OpenAlex ID: W2053811523
arXiv ID: 1108.2969
DOI: 10.1112/BLMS/BDS026
review text: Let \(Y\times\mathbb{R}\) be the symplectization of a contact manifold \(Y\) and \(L\subset Y\times\mathbb{R}\) a Lagrangian submanifold. The author calls a transverse intersection \(L\cap Y\times\left\{t\right\}\) collarable if it can be isotoped through Lagrangian submanifolds to a cylinder over a Legendrian embedding near \(Y\times\left\{t\right\}\). Such a notion arises naturally (via the trivialization through the Liouville vector field) when studying intersections of Lagrangian submanifolds with contact hypersurfaces in sympectic manifolds.NEWLINENEWLINEThe paper under review gives two explicit examples of Lagrangian disks in \(\mathbb C^{2}\) transverse to \(S^{3}\) whose slices are non-collarable. The one example is a Lagrangian cobordism T from the trivial Legendrian knot \(K_0\subset S^3\) to the empty set with one global maximum \(m\) for the `distance from the origin' function such that, for any sufficiently small \(\delta > 0\), the slice \(T \cap S^{3}_{m-\delta}\) is non-collarable. The other example yields Lagrangian disks in \(D^4\) transverse to \(S^3\) whose intersections with \(S^3\) are non-collarable.
reviewed by: Thilo Kuessner
description: scientific article; zbMATH DE number 6092629