Flexibility of the Hamiltonian adjoint action and classification of bi-invariant metrics

Abstract

On an open, connected symplectic manifold $(M,\omega )$, the group of Hamiltonian diffeomorphisms forms an infinite-dimensional Fréchet Lie group with Lie algebra $C_c^{\infty }(M)$ and adjoint action given by pullbacks. We prove that this action is flexible: for any non-constant $u\in C^{\infty }(M)$, every $f\in C_c^{\infty }(M)$ can be expressed as a weighted finite sum of elements from the adjoint orbit of $u$, with total weight bounded by constant multiple of $\Vert f{\Vert }_{\infty }+\Vert f{\Vert }_{L^1}$. Consequently, all $\mathrm{Ham}(M,\omega )$-invariant norms on $C_c^{\infty }(M)$ are dominated by a sum of $L^{\infty }$ and $L^1$ norms. As an application, we classify up to equivalence all bi-invariant pseudo-metrics on the group of Hamiltonian diffeomorphisms of an exact symplectic manifold, answering a question of Eliashberg and Polterovich.