Recently, L. Buhovsky, A. Logunov and S. Tanny proved the (strong) Poisson
bracket conjecture by Leonid Polterovich in dimension 2. In this note, instead
of open cover constituted of displaceable sets in their work, considering open
cover constituted of topological discs we give a sufficient and necessary
condition that Poisson bracket invariants of these covers are positive.Comment: 10 page

Lu, Guangcun

Shi, Kun

English

arXiv

L. Buhovsky, A. Logunov and S. Tanny proved the (strong) Poisson bracket
conjecture by Leonid Polterovich in dimension $2$. In this note, instead of
open cover consisting of displaceable sets in their work, we consider open
cover constituted of topological discs and give a necessary and sufficient
condition that Poisson bracket invariants of these covers are positive.Comment: v3: 18 pages, new title, accepted for publication in Tokyo Journal of
  Mathematic

Lower bound for the Poisson bracket invariant on surfaces

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