Abstract
We prove the Poisson version of the Eliashberg-Gromov \(C^0\)-rigidity. More precisely, we prove that the group of Poisson diffeomorphisms is closed with respect to the \(C^0\) topology inside the group of all diffeomorphisms. The proof relies on the Poisson version of the energy-capacity inequality.
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Notes
Note that the Leibniz identity (1) implies that \(\{f_t, \cdot \}\) is a derivation of \(C^{\infty }(M)\) and as such it defines a (time-dependent) vector field on M.
Note that for symplectic manifolds, the closedness of the symplectic form is equivalent with the Jacobi identity for the corresponding Poisson bracket.
Compact and without boundary.
Here by section we mean with respect to the target map, i.e. \(t\circ b = {\text {id}}_M: M \rightarrow M\).
Actually, in [21] the non-degeneracy of the Hofer norm was claimed for regular Poisson manifolds, but in the proof they do not use regularity, but the assumption that the restriction of a compactly supported function to a leaf is compactly supported, however, without stating this explicitly.
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Acknowledgements
I would like to thank Ioan Mărcuț for carefully reading and providing a very useful feedback on a preliminary version of the article and for suggesting Questions 1 and 3, and to Marius Crainic for an interesting discussion and suggesting Remark 2. The idea for the proof of Theorem 1 arose while I was preparing a talk for the UGC seminar at Utrecht University and therefore I would like to thank Fabian Ziltener and Álvaro del Pino Gómez for inviting me to give a talk. The work on this project was funded by Agence Nationale de la Recherche through “ANR COSY: New challenges in contact and symplectic topology” grant (decision ANR-21-CE40-0002).
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Joksimović, D. \(C^0\)-rigidity of Poisson diffeomorphisms. Lett Math Phys 113, 69 (2023). https://doi.org/10.1007/s11005-023-01696-6
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DOI: https://doi.org/10.1007/s11005-023-01696-6