Bott-integrable contact forms with large systolic ratio

2026/04/21 by Hansjörg Geiges, Jakob Hedicke, Murat Sağlam · 1 voice
Mathematics · #math.SG #math.DS

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Abstract

We show that there is no universal upper bound for the systolic ratio of Bott-integrable contact forms on closed 3-manifolds, thus providing further evidence for the relative flexibility of integrable contact forms. For the proof, we study piecewise linear approximations of Lutz forms and establish integrability of a `plug' constructed by Abbondandolo, Bramham, Hryniewicz and Salomão for pushing up the systolic ratio.

Citations

• Bott-integrability of overtwisted contact structures
• Systolic inequalities for S1-invariant contact forms in dimension three
• Sharp systolic inequalities for invariant tight contact forms on principal S1-bundles over S2
• Symplectic capacities of domains close to the ball and Banach-Mazur geodesics in the space of contact forms
• A gap in the Hofer metric between integrable and autonomous Hamiltonian diffeomorphisms on surfaces
• C⁰-gap between entropy-zero Hamiltonians and autonomous diffeomorphisms of surfaces
• Reeb flows with small contact volume and large return time to a global section
• Contact forms with large systolic ratio in arbitrary dimensions
• Finsler geodesics, periodic Reeb orbits, and open books
• Sharp systolic inequalities for Reeb flows on the three-sphere
• Contact geometry and isosystolic inequalities
• The Seiberg–Witten equations and the Weinstein conjecture

Discussions

• Hansj\"org Geiges, Jakob Hedicke, Murat Sa\u{g}lam: Bott-integrable contact forms with large systolic ratio https://arxiv.org/abs/2604.19237 https://arxiv.org/pdf/2604.19237 https://arxiv.org/html/260 [bsky, 0 points, 0 comments]

Related

• Contact forms with large systolic ratio in arbitrary dimensions
• Bott-integrability of overtwisted contact structures
• Systolic inequalities for S1-invariant contact forms in dimension three
• A local contact systolic inequality in dimension three
• Reeb flows with small contact volume and large return time to a global section