Journal of Symplectic Geometry

Volume 17 (2019)

Number 5

Moser–Greene–Shiohama stability for families

Pages: 1427 – 1446



Álvaro Pelayo (Department of Mathematics, University of California at San Diego)

Xiudi Tang (Department of Mathematical and Computational Sciences, University of Toronto, Mississauga, Ontario, Canada)


Let $M$ be a noncompact oriented connected manifold and let $B$ be a compact manifold. We give conditions on two smooth families of volume forms ${\lbrace \omega_p \rbrace}_{p \in B} , {\lbrace \tau_p \rbrace}_{p \in B}$ which guarantee the existence of a smooth family of diffeomorphisms ${\lbrace \varphi_p \rbrace}_{p \in B}$ such that $\varphi^{\ast}_p \omega_p = \tau_p$ for all $p \in B$. If $B$ is a point, our result recovers a theorem of Greene and Shiohama from 1979, which itself extended a theorem of Moser for compact manifolds.

Received 5 February 2017

Accepted 5 September 2018

Published 20 November 2019