Symplectic convexity in low-dimensional topology
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 versions are equivalent and implied by \(\omega\)-complexity. In dimensions greater than 4, \(\omega\)-convexity is equivalent to dominating which implies both, pseudo-convexity and weak symplectic filling; pseudo-convexity also implies weak symplectic filling. The question of whether weak symplectic filling implies pseudo-convexity in dimensions greater than 4 is open; other unmentioned implications do not hold. Since the weaker versions of convexity are easier to show, the question is posed as to what conditions have to be added to them to imply \(\omega\)-convexity.
scientific article; zbMATH DE number 1307947
scientific article