Positive loops and \(L^{\infty }\)-contact systolic inequalities

Abstract

We prove an inequality between the \(L^{\infty }\)-norm of the contact Hamiltonian of a positive loop of contactomorphims and the minimal Reeb period. This implies that there are no small positive loops on hypertight or Liouville fillable contact manifolds. Non-existence of small positive loops for overtwisted 3-manifolds was proved by Casals et al. (J Symplectic Geom 14:1013–1031, 2016). As corollaries of the inequality we deduce various results. E.g. we prove that certain periodic Reeb flows are the unique minimisers of the \(L^{\infty }\)-norm. Moreover, we establish \(L^\infty \)-type contact systolic inequalities in the presence of a positive loop.

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Albers, P., Fuchs, U. & Merry, W.J. Positive loops and \(L^{\infty }\)-contact systolic inequalities. Sel. Math. New Ser. 23, 2491–2521 (2017). https://doi.org/10.1007/s00029-017-0338-2

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Mathematics Subject Classification

  • 57R58
  • 53D10