Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 2

Special issue in honor of Joseph J. Kohn on the occasion of his 90th birthday

Guest Editors: J.E. Fornaess, Stanislaw Janeczko, Duong H. Phong, and Stephen S.T. Yau

On the adjoint action of the group of symplectic diffeomorphisms

Pages: 657 – 682

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n2.a14


László Lempert (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)


We study the action of Hamiltonian diffeomorphisms of a compact symplectic manifold $(X, \omega)$ on $C^\infty (X)$ and on functions $C^\infty (X) \to \mathbb{R}$. We describe various properties of invariant convex functions on $C^\infty (X)$. Among other things we show that continuous convex functions $C^\infty (X) \to \mathbb{R}$ that are invariant under the action are automatically invariant under so called strict rearrangements and they are continuous in the sup norm topology of $C^\infty (X)$; but this is not generally true if the convexity condition is dropped.

The author’s research was partially supported by NSF grant DMS 1764167.

Received 8 January 2021

Received revised 2 September 2021

Accepted 29 September 2021

Published 13 May 2022