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arXiv · 1602.01383

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

Peter Albers·Urs Fuchs·Will J. Merry
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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-Presas-Sandon in [CPS16]. As corollaries of the inequality we deduce various results. E.g. we prove that certain periodic Reeb flows are the unique minimizers 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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Peter Albers, Urs Fuchs, Will J. Merry. 2016-02-03. Positive loops and $L^{\infty}$-contact systolic inequalities. https://doi.org/10.1007/s00029-017-0338-2

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