Bi-invariant metrics on the group of symplectomorphisms

Abstract:

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