Positive loops and L∞-contact systolic inequalities - Albers, Peter; Fuchs, Urs; Merry, Will J. (Selecta mathematica, 2017)
We prove an inequality between the L∞-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∞-norm. Moreover, we establish L∞-type contact systolic inequalities in the presence of a positive loop.