Bi-invariant Metrics on the Group of Symplectomorphisms

This paper studies the extension of the Hofer metric and general Finsler metrics on the Hamiltonian symplectomorphism group Ham(M,ω) to the identity component Symp0(M,ω) of the symplectomorphism group. In particular, we prove that the Hofer metric on Ham(M,ω) does not extend to a bi-invariant metric on Symp0(M,ω) for many symplectic manifolds. We also show that for the torus T2n with the standard symplectic form ω, no Finsler metric on Ham(T2n, ω) that satisfies a strong form of the invariance condition can extend to a bi-invariant metric on Symp0(T 2n, ω). Another interesting result is that there exists no C1-continuous bi-invariant metric on Symp0(T 2n, ω).