Removing a Ray from a Noncompact Symplectic Manifold

Abstract

We prove that any noncompact symplectic manifold which admits a properly embedded ray with a wide neighborhood is symplectomorphic to the complement of the ray by constructing an explicit symplectomorphism in the case of the standard Euclidean space. We use this excision trick to construct a nowhere vanishing Liouville vector fields on every cotangent bundle. γ 0 (s) = (0, . . . , 0, s, 0), arXiv:1812.00444v2 [math.SG]

Symplectic invariants of semitoric systems and the inverse problem for quantum systems

“…There is also recent work on the so called toric-focus systems [98] and systems which have semitoric features but also include singularities with hyperbolic components [39]. In the article [104] there is an application of the ideas in the proof of this classification to study symplectic forms on noncompact manifolds.…”

Removing parametrized rays symplectically

“…[Gro85,McD87,MT93,Tra93]. More recently, X. Tang showed that for a general manifold M the subset N can be chosen to be a ray if the ray possesses a "wide neighborhood" ( [Tan20]). Roughly speaking a ray is a 2-ended connected non-compact 1dimensional local submanifold whose one end closes up inside M while at the other end the embedding is proper.…”

“…In this paper an extension to higher dimensional sets regarded as parametrized rays is provided. While for those higher dimensional sets a condition is needed, this condition is trivially fulfilled for an isolated ray as treated in [Tan20].…”