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 $a$.