Hofer-Like Geometry Revisited

Résumé

We prove that the inclusion of the Hamiltonian group $\Ham(M,ω)$ into the identity component \(G_ω(M)\) of the symplectic diffeomorphism group is a bi-Lipschitz embedding with respect to the Hofer norm and the Hofer-like norm, and we identify geometric conditions under which this embedding is isometric: settling a conjecture of Banyaga. This conjecture was proved by Buss and Leclercq; our proof provides explicit equivalence constants. We also detail and simplify Banyaga's original proof of the non-degeneracy of the Hofer-like norm. We then extend the analysis to all of \(G_ω(M)\): for \(ϕ\) with flux class \(γ\), the Hofer-like norm is given by a semidirect-product formula, the infimum over the harmonic locus \(\Harm(γ)\) plus a Hofer residue. This yields a geometric condition for the two norms to agree on the Hamiltonian group. In particular, this geometric condition holds on all closed surfaces of genus $g\ge 2$.

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