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

TL;DR

This paper establishes an inequality linking the contact Hamiltonian of positive loops to the minimal Reeb period, leading to new insights on the non-existence of small positive loops in certain contact manifolds and related systolic inequalities.

Contribution

It introduces a novel inequality connecting contact Hamiltonians and Reeb periods, and applies it to prove non-existence results and systolic inequalities in contact geometry.

Findings

No small positive loops on hypertight or Liouville fillable contact manifolds.

Certain periodic Reeb flows are the unique minimizers of the $L^ abla$-norm.

Established $L^{ abla}$-type contact systolic inequalities with positive loops.

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.