On the Uniqueness of Hofer's Geometry

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.