$C^0$-rigidity of Poisson diffeomorphisms

TL;DR

This paper establishes that Poisson diffeomorphisms are $C^0$-rigid, meaning they form a closed subgroup within all diffeomorphisms, by extending the Gromov-Eliashberg rigidity to the Poisson setting.

Contribution

It proves the $C^0$-rigidity of Poisson diffeomorphisms, a significant extension of known rigidity results to the Poisson geometric context.

Findings

Poisson diffeomorphisms form a closed subgroup in the $C^0$ topology.

The proof uses a Poisson version of the energy-capacity inequality.

The result generalizes Gromov-Eliashberg's $C^0$-rigidity to Poisson manifolds.

Abstract

We prove the Poisson version of the Gromov-Eliashberg's $C^0$-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.