On the extremality of Hofer's metric on the group of Hamiltonian diffeomorphisms

Abstract

Let M be a closed symplectic manifold, and let k · k be a norm on the space of all smooth functions on M, which are zero-mean normalized with respect to the canonical volume form. We show that if k· k ≤ Ck · k∞, and k · k is invariant under the action of Hamiltonian diffeomorphisms, then it is also invariant under all volume preserving diffeomorphisms. We also prove that if k·k is, additionally, not equivalent to k · k∞, then the induced Finsler metric on the group Ham(M, !) of Hamiltonian diffeomorphisms on M vanishes identically. These results provide partial answers to questions raised by Eliashberg and Polterovich in [3]. Both results rely on an extension of k · k to the space of essentially bounded measurable functions, which is invariant under all measure preserving bijections.