Examples of symplectic non-leaves

TL;DR

This paper investigates which manifolds can serve as leaves in codimension-1 symplectic foliations, introducing new examples and conditions related to strongly geometrically bounded symplectic forms and their realizability.

Contribution

It demonstrates that certain manifolds admit strongly geometrically bounded symplectic forms and explores their potential to be realized as symplectic leaves, providing new examples and obstructions.

Findings

Leaves of symplectic foliations are strongly geometrically bounded.

Manifolds can admit strongly geometrically bounded forms but not be realized as leaves.

The blowup of Euclidean space at infinitely many points has complex leaf realizability properties.

Abstract

This paper deals with the following question: which manifolds can be realized as leaves of codimension-1 symplectic foliations on closed manifolds? We first observe that leaves of symplectic foliations are necessarily strongly geometrically bounded. We show that a symplectic structure which admits an exhaustion by compacts with (convex) contact boundary can be deformed to a strongly geometrically bounded one. We then give examples of smooth manifolds which admit a strongly geometrically bounded symplectic form and can be realized as a smooth leaf, but not as a symplectic leaf for any choice of symplectic form on them. Lastly, we show that the (complex) blowup of 2n-dimensional Euclidean space at infinitely many points, both admits strongly geometrically bounded symplectic forms for which it can and cannot be realized as a symplectic leaf.