On the extremality of Hofer's metric on the group of Hamiltonian diffeomorphisms

2005/01/10 by Yaron Ostrover, Roy Wagner, Ostrover, Yaron +1 · 1 citation
Mathematics · #46B99 #53D05 #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Symplectic Geometry (math.SG) #math.FA #math.SG #msc:46B99 #msc:53D05

paper · pdf · doi:10.48550/arxiv.math/0501143

Latex, 17 pages

arxiv created 2005/01/10 · openalex publication_date 2005/01/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be a closed symplectic manifold, and let | | be a norm on the space of all smooth functions on M, which are zero-mean normalized with respect to the canonical volume form. We show that if | | is dominated from above by the L-Infinity-norm, and | | is invariant under the action of Hamiltonian diffeomorphisms, then it is also invariant under all volume preserving diffeomorphisms. We also prove that if | | is, additionally, not equivalent to the L-Infinity-norm, then the induced Finsler metric on the group of Hamiltonian diffeomorphisms on M vanishes identically.

Cited by

• Flexibility of the Hamiltonian adjoint action and classification of bi-invariant metrics

Related

• On the Uniqueness of Hofer's Geometry
• A Hofer-like metric on the group of symplectic diffeomorphisms
• A Comparison of Hofer's Metrics on Hamiltonian Diffeomorphisms and Lagrangian Submanifolds
• Bi-invariant metrics on the group of symplectomorphisms
• A Comparison between Hofer's metric and C¹-topology