Systolic inequalities for 𝕊¹-invariant contact forms in dimension three

In contact geometry, a systolic inequality is a uniform upper bound on the shortest period of a closed Reeb orbit, in terms of the contact volume. We prove a general systolic inequality valid on Seifert bundles with non-zero Euler number for all contact forms that are invariant under the underlying circle action. This is essentially optimal in the sense that invariance under a finite group action is not enough to force the existence of a systolic inequality.