Existence of pseudoheavy fibers of moment maps

Morimichi Kawasaki, Ryuma Orita
Communications in Contemporary Mathematics, vol. 23 (2021), no. 5, 2050047
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Overview

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A moment map formed from Poisson-commuting functions organizes a symplectic manifold into their common level sets, or fibers. Entov and Polterovich proved that, on a closed symplectic manifold, at least one fiber cannot be displaced from itself by a Hamiltonian diffeomorphism. They also asked whether one can find a fiber satisfying the stronger condition of heaviness.

We introduce pseudoheaviness, a weaker condition than heaviness that still implies non-displaceability. For every partial symplectic quasi-state on a closed symplectic manifold, we prove that every moment map formed from finitely many Poisson-commuting functions has a pseudoheavy fiber. When the quasi-state is simple, this fiber is heavy, giving a partial answer to their question. Applications also produce a family of singular Lagrangian submanifolds in $S^2\times S^2$ with different displacement properties relative to other Lagrangian submanifolds.

@article{kawasaki2021existence,
  author        = {Kawasaki, Morimichi and Orita, Ryuma},
  title         = {{Existence of pseudoheavy fibers of moment maps}},
  journal       = {Communications in Contemporary Mathematics},
  volume        = {23},
  number        = {5},
  pages         = {2050047},
  year          = {2021},
  doi           = {10.1142/S0219199720500479},
  eprint        = {1901.09395},
  archivePrefix = {arXiv}
}