Sharp systolic inequalities for invariant tight contact forms on principal S1-bundles over S2

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 form 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.