The $g$-areas and the commutator length

Abstract

The commutator length of a Hamiltonian diffeomorphism $f\in \mathrm{Ham}(M, \omega)$ of a closed symplectic manifold $(M,\omega)$ is by definition the minimal $k$ such that $f$ can be written as a product of $k$ commutators in $\mathrm{Ham}(M, \omega)$. We introduce a new invariant for Hamiltonian diffeomorphisms, called the $k_+$-area, which measures the "distance", in a certain sense, to the subspace $\mathcal{C}_k$ of all products of $k$ commutators. Therefore this invariant can be seen as the obstruction to writing a given Hamiltonian diffeomorphism as a product of $k$ commutators. We also consider an infinitesimal version of the commutator problem: what is the obstruction to writing a Hamiltonian vector field as a linear combination of $k$ Lie brackets of Hamiltonian vector fields? A natural problem related to this question is to describe explicitly, for every fixed $k$, the set of linear combinations of $k$ such Lie brackets. The problem can be obviously reformulated in terms of Hamiltonians and Poisson brackets. For a given Morse function $f$ on a symplectic Riemann surface $M$ (verifying a weak genericity condition) we describe the linear space of commutators of the form $\{f,g\}$, with $g\in\mathcal{C}^\infty(M,\mathbb{R})$.