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

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

2005-01-10

arxiv.org/abs/math/0501143v1

Mathematics

Symplectic Geometry

Latex, 17 pages

Scientific paper

Let M be a closed symplectic manifold, and let | | 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 | | is dominated from above by the L-Infinity-norm, and | | is invariant under the action of Hamiltonian diffeomorphisms, then it is also invariant under all volume preserving diffeomorphisms. We also prove that if | | is, additionally, not equivalent to the L-Infinity-norm, then the induced Finsler metric on the group of Hamiltonian diffeomorphisms on M vanishes identically.

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