Symplectic convexity in low-dimensional topology
type: Item
instance of: scholarly article
title: Symplectic convexity in low-dimensional topology
zbMATH Open document ID: 0930.53049
DOI: 10.1016/S0166-8641(97)00196-X
author: John B. Etnyre
published in: Topology and its Applications
publication date: 2000-02-15T00:00:00Z
review text: If \((X,\omega)\) is a symplectic manifold and \(S\) a compact hypersurface, then \(S\) has contact type if there is symplectic dilation \(\nu\) in a neighborhood of \(S\) that is transverse to \(S\), i.e., a vector field such that \(L_\nu\omega=\omega\) (after normalizing). Such a hypersurface inherits a co-oriented contact structure (there is a nonzero vector field transverse to the contact fields). Conversely, if \((S,\xi)\) is a co-oriented contact manifold, then there is a symplectic manifold in which \(S\) sits as a hypersurface of contact type. A domain \(U\) in \((X,\omega)\) bounded by a hypersurface \(S\) is \(\omega\)-convex if there is a vector field \(\nu\), defined in a neighborhood of \(S\), that is transverse to \(S\), points out of \(U\), and in a symplectic dilation -- thus \(S\) is a hypersurface of contact type and the symplectic dilation points out of \(U\). This paper studies various versions of convexity in symplectic geometry starting with \(\omega\)-convexity, the strongest. Examples are given of constructions of \(\omega\)-convex hypersurfaces and various constructions of symplectic manifolds are interpreted in terms of \(\omega\)-convexity. \(\omega\)-convexity is a necessary condition for most cut-and-paste constructions of symplectic manifolds. Three weaker versions of convexity are then explored, weakly symplectically fillable, dominating, and pseudo convex. \((M,\xi)\), a co-oriented contact manifold, is symplectically fillable if there is a compact symplectic manifold \((X,\omega)\) with \(M\) as the oriented boundary and \(\omega|_\xi\) nondegenerate. If \((X,\omega)\) is a symplectic manifold with \(\partial X=M\), the symplectic form \(\omega\) dominates the contact structure \(\xi\) if \(\omega|_\xi\) is the canonical conformal class of symplectic forms on \(\xi\). Pseudo-convexity is defined in terms of an almost complex structure on \(X\) which can be related to a symplectic form \(\omega\). In dimension 4, the three weaker vers…
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1307947
zbMATH Keywords: \(\omega\)-convexity
zbMATH Keywords: symplectic manifold
zbMATH Keywords: cut-and-paste constructions
zbMATH Keywords: symplectically fillable
zbMATH Keywords: pseudoconvexity
zbMATH Keywords: contact manifold
reviewed by: George E. Lang jun.
MaRDI profile type: Publication
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full work available at URL: s0166-8641(97)00196-x
OpenAlex ID: W1987655880
description: scientific article; zbMATH DE number 1307947