The Hofer norm of a contactomorphism

TL;DR

This paper introduces a new non-degenerate, right-invariant metric on the contactomorphism group derived from the contact Hamiltonian's L-infinity norm, exploring its properties, relations to existing metrics, and implications for contact geometry.

Contribution

It defines a novel contact Hofer metric, analyzes its non-degeneracy, invariance properties, and links to Hofer's metric and Sandon's conjecture in contact geometry.

Findings

The contact Hofer metric is non-degenerate and depends on the contact form.

It has infinite diameter in several cases.

The metric relates to the Reeb flow and supports a form of Sandon's conjecture.

Abstract

We show that the $L^{\infty}$-norm of the contact Hamiltonian induces a non-degenerate right-invariant metric on the group of contactomorphisms of any closed contact manifold. This contact Hofer metric is not left-invariant, but rather depends naturally on the choice of a contact form $\alpha,$ whence its restriction to the subgroup of $\alpha$-strict contactomorphisms is bi-invariant. The non-degeneracy of this metric follows from an analogue of the energy-capacity inequality. We show furthermore that this metric has infinite diameter in a number of cases by investigating its relations to previously defined metrics on the group of contact diffeomorphisms. We study its relation to Hofer's metric on the group of Hamiltonian diffeomorphisms, in the case of prequantization spaces. We further consider the distance in this metric to the Reeb one-parameter subgroup, which yields an intrinsic formulation of a small-energy case of Sandon's conjecture on the translated points of a contactomorphism. We prove this Chekanov-type statement for contact manifolds admitting a strong exact filling.