On the non-existence of small positive loops of contactomorphisms on overtwisted contact manifolds

TL;DR

This paper proves that overtwisted contact manifolds cannot have positive loops of contactomorphisms generated by small Hamiltonian functions, highlighting a fundamental restriction in contact topology.

Contribution

It establishes a non-existence result for small positive loops of contactomorphisms on overtwisted contact manifolds, a new insight in contact geometry.

Findings

No positive loops generated by $C^0$-small Hamiltonians on overtwisted manifolds

Restrictions on contactomorphism dynamics in overtwisted settings

Advances understanding of contactomorphism group structure

Abstract

We prove that on overtwisted contact manifolds there can be no positive loops of contactomorphisms that are generated by a $C^0$-small Hamiltonian function.