Volume 8 (2025)
Vialaret, Simon
Sharp systolic inequalities for invariant tight contact forms on principal $\mathbb{S}^1$-bundles over $\mathbb{S}^2$
Annales Henri Lebesgue, Volume 8 (2025), pp. 995-1021

Metadata

DOI10.5802/ahl.253
Keywords Contact dynamics ,  systolic inequalities ,  surfaces of section

Abstract

The systole of a contact form $\alpha $ is defined as the shortest period of closed Reeb orbits of $\alpha $. Given a non-trivial $\mathbb{S}^1$-principal bundle over $\mathbb{S}^2$ with total space $M$, we prove a sharp systolic inequality for the class of tight contact forms on $M$ invariant under the $\mathbb{S}^1$-action. This inequality exhibits a behavior which depends on the Euler class of the bundle in a subtle way. As applications, we prove a sharp systolic inequality for rotationally symmetric Finsler metrics on $\mathbb{S}^2$, a systolic inequality for the shortest contractible closed Reeb orbit, and a particular case of a conjecture by Viterbo.


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