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arXiv · 2403.02228

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

Simon Vialaret
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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.

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Simon Vialaret. 2024-03-04. Sharp systolic inequalities for invariant tight contact forms on principal S1-bundles over S2. https://arxiv.org/abs/2403.02228

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