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 2 × 2 . Polterovich and Shelukhin have recently constructed a family of functionals on the space of time-dependent Hamiltonian functions on S 2 (1) × S 2 (a) for rational 0 < a < 1 , called Lagrangian spectral estimators. Using our fragmentation result we prove that the restriction of their functionals to the subdomain 2 (c) × 2 (d) is a uniformly C 0 -continuous functional where 0 < c , d < 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 2 (c) × 2 (d) 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 parameters c and  d. [ABSTRACT FROM AUTHOR]