Hamiltonian fragmentation in dimension four with application to spectral estimators

We prove a new Hamiltonian extension and consequently a fragmentation result in dimension $4$ for the symplectic manifold ${\mathbb{D}}^2\times {\mathbb{D}}^2$. Polterovich and Shelukhin have recently constructed a family of functionals on the space of time dependent Hamiltonian functions on $S^2(1)\times S^2(a)$ for certain rational $0<a<1$, called Lagrangian spectral estimators. Using our fragmentation result we prove that the restriction of their functionals to the subdomain ${\mathbb{D}}^2(c)\times {\mathbb{D}}^2(a)$ is a uniformly $C^0$-continuous functional where $0<c<1$. As an application of our results, we show that the complement of a Hofer ball in the group of compactly supported Hamiltonian diffeomorphisms of ${\mathbb{D}}^2(c)\times {\mathbb{D}}^2(a)$ contains a $C^0$-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 values of parameters $c$ and $a$.