Examples of symplectic non-leaves

Abstract

This paper deals with the following question: which manifolds can be realized as leaves of codimension-$1$ symplectic foliations (of regularity at least $C^2$) 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 admits both strongly geometrically bounded symplectic forms for which it can and cannot be realized as a symplectic leaf.