Sharp systolic inequalities for invariant tight contact forms on principal S1-bundles over S2 · arXivDesk
2403.02228Mar 4, 202423 pages; v3: minor corrections; v2: corrected a mistake in the proof of the main theorem, new section 4.2 on a sharp bound on the systolic ratio for contractible Reeb orbits
Sharp systolic inequalities for invariant tight contact forms on principal S1-bundles over S2
The systole of a contact form α is defined as the shortest period of closed Reeb orbits of α. Given a non-trivial S1-principal bundle over S2 with total space M
Nearby in the stack
, we prove a sharp systolic inequality for the class of tight contact form on
M
invariant under the
S1
-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
S2
, a systolic inequality for the shortest contractible closed Reeb orbit, and a particular case of a conjecture by Viterbo.