Hamiltonian fragmentation in dimension four with application to spectral estimators

Abstract

We prove a new Hamiltonian extension and consequently a fragmentation result in dimension [Formula: see text] for the symplectic manifold [Formula: see text]. Polterovich and Shelukhin have recently constructed a family of functionals on the space of time-dependent Hamiltonian functions on [Formula: see text] for rational [Formula: see text], called Lagrangian spectral estimators. Using our fragmentation result we prove that the restriction of their functionals to the subdomain [Formula: see text] is a uniformly [Formula: see text]-continuous functional where [Formula: see text]. As an application of our results, we show that the complement of a Hofer ball in the group of compactly supported Hamiltonian diffeomorphisms of [Formula: see text] contains a [Formula: see text]-open subset. Finally, we show that the aforementioned group equipped with the Hofer distance admits an isometric embedding of an infinite-dimensional flat space for suitable parameters [Formula: see text] and [Formula: see text].