Examples of symplectic non-leaves

• Published in: Revista matemática iberoamericana Vol. 41; no. 3; pp. 1081 - 1100
• Main Authors: Gironella, Fabio, Toussaint, Lauran
• Format: Journal Article
• Language: English
• Published: European Mathematical Society 06.01.2025
• Subjects: Mathematics; Symplectic Geometry; symplectic; Mathematics Subject Classification 2020: 57R17 (primary); 57R30 (secondary) foliation
• ISSN: 0213-2230; 2235-0616
• DOI: 10.4171/rmi/1520

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.