Invariant Finsler metrics on the space of Lagrangian embeddings (Q1579100)

Let \(L\) be a closed embedded Lagrangian submanifold of a symplectic manifold \((P,\omega)\). \({\mathcal H}(P)\) is the space of compactly supported functions on \([0,1]\times P\). Any \(H\in{\mathcal H}(P)\) defines a time-dependent Hamiltonian flow on \(P\). Time one maps of such flows form a group \({\mathcal S}={\mathcal S}(P,\omega)\) called the group of Hamiltonian symplectomorphisms of \(P\). A Lagrangian manifold \(L'\in P\) is called Hamiltonian isotopic to \(L\) if \(L'= g(L)\) for some \(g\in{\mathcal S}\). Let \({\mathcal L}={\mathcal L}(P,\omega, L)\) be the space of such manifolds; the group \({\mathcal S}\) acts naturally on \({\mathcal L}\). This paper deals with \({\mathcal S}\)-invariant metrics on \({\mathcal L}\). The author shows that the Hofer metric \(\delta_H\) is non-degenerate when \(P\) is geometrically bounded, and every \({\mathcal S}\)-invariant Finsler metric on \({\mathcal L}\) is a multiple of \(\delta_H\) if \(\dim{\mathcal L}\geq 1\). The author also discusses \({\mathcal S}\)-invariant Finsler metrics on the space \({\mathcal L}^*\) of Lagrangian submanifolds isotopic to \({\mathcal L}\) in the class of Lagrangian embeddings. It turns out that all \({\mathcal S}\)-invariant Finsler metrics on \({\mathcal L}^*\) are highly degenerate provided that \(\dim H^1({\mathcal L},R)\geq 1\) and \(\dim L\geq 1\).