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 ${\rm Ham}(M,\omega)$ to the identity component ${\rm Symp}_0(M,\omega)$ of the symplectomorphism group. In particular, we prove that the Hofer metric on ${\rm Ham}(M,\omega)$ does not extend to a bi-invariant metric on ${\rm Symp}_0(M,\omega)$ for many symplectic manifolds. We also show that for the torus $\mathbb{T}^{2n}$ with the standard symplectic form $\omega$, no Finsler metric on ${\rm Ham}(\mathbb{T}^{2n},\omega)$ that satisfies a strong form of the invariance condition can extend to a bi-invariant metric on ${\rm Symp}_0(\mathbb{T}^{2n},\omega)$. Another interesting result is that there exists no $C^1$-continuous bi-invariant metric on ${\rm Symp}_0(\mathbb{T}^{2n},\omega)$.