On the Uniqueness of Hofer's Geometry

TL;DR

This paper proves that any continuous, Hamiltonian-invariant norm on smooth functions on a closed symplectic manifold is dominated by the supremum norm, leading to the uniqueness of Hofer's metric among such bi-invariant Finsler metrics.

Contribution

It establishes the uniqueness of Hofer's metric by showing all continuous, Hamiltonian-invariant norms induce either trivial or equivalent metrics.

Findings

Any such norm is dominated by the $L_{}$-norm.

Bi-invariant Finsler metrics generated by these norms are either zero or equivalent to Hofer's metric.

The result characterizes the rigidity of Hamiltonian diffeomorphism metrics.

Abstract

We study the class of norms on the space of smooth functions on a closed symplectic manifold, which are invariant under the action of the group of Hamiltonian diffeomorphisms. Our main result shows that any such norm that is continuous with respect to the $C^{\infty}$-topology, is dominated from above by the $L_{\infty}$-norm. As a corollary, we obtain that any bi-invariant Finsler pseudo-metric on the group of Hamiltonian diffeomorphisms that is generated by an invariant norm that satisfies the aforementioned continuity assumption, is either identically zero or equivalent to Hofer's metric.