Reeb flows with small contact volume and large return time to a global section
Abstract
We show that any co-oriented closed contact manifold of dimension at least five admits a contact form such that the contact volume is arbitrarily small but the Reeb flow admits a global hypersurface of section with the property that the minimal period on the boundary of the hypersurface and the first return time in the interior of the hypersurface are bounded below. An immediate consequence of this statement is that every co-oriented contact structure on any closed manifold admits a contact form with arbitrarily large systolic ratio. This generalizes the result of Abbondandolo et al. in dimension three to higher dimensions. The proof the main result is inductive and uses the result of Abbondandolo et al. on large systolic ratio in dimension three in its basis step. The essential construction in the proof relies on the Giroux correspondence in higher dimensions.
Keywords
Cite
@article{arxiv.2003.04729,
title = {Reeb flows with small contact volume and large return time to a global section},
author = {Murat Sağlam},
journal= {arXiv preprint arXiv:2003.04729},
year = {2021}
}
Comments
16 pages. The main result in this version is a novel statement that implies the main result of the previous version. The title, the abstract and the introduction are changed in parallel to the main result of this version. The essential construction in the proof is a slight modification of the proof of the previous main result. arXiv admin note: substantial text overlap with arXiv:1806.01967