The Hofer norm of a contactomorphism

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.

Keywords

contact Hamiltonian, right-invariant metric, contactomorphism group, translated points, Hofer norm

2010 Mathematics Subject Classification

37J55, 53D10, 57R17, 57S05

Full Text (PDF format)

I thank Sheila Margherita Sandon and Leonid Polterovich for introducing me to the main subjects of this paper over the course of numerous conversations. I thank Strom Borman, Viktor Ginzburg, Michael Khanevsky, Boris Khesin, Fran¸cois Lalonde, Dmitry Tonkonog, Michael Usher and Frol Zapolsky for useful conversations. I thank Strom Borman, Leonid Polterovich, Sheila Margherita Sandon and Frol Zapolsky for their comments on an earlier version of this manuscript. The work on this paper has started right after the workshop “Rigidity and Flexibility in Symplectic Topology and Dynamics” at the Lorentz Center (Leiden, 2014). I thank its organizers and participants for a very enjoyable and stimulating event. This work was carried out at CRM, University of Montreal, and I thank this institution for its warm hospitality. I thank the referees for their very careful reading and useful suggestions.

Received 20 September 2015

Accepted 15 September 2016

Published 28 November 2017