On the Uniqueness of Hofer's Geometry

2010/08/03 by Buhovsky, Lev, Ostrover, Yaron · 1 citation
#22E65 #53D05 #58B20 #FOS: Mathematics #Functional Analysis (math.FA) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1008.0661

Abstract

We study the class of norms on the space of smooth functions on a closed symplectic manifold, which are invariant under the action of the group of Hamiltonian diffeomorphisms. Our main result shows that any such norm that is continuous with respect to the C^∞-topology, is dominated from above by the L_∞-norm. As a corollary, we obtain that any bi-invariant Finsler pseudo-metric on the group of Hamiltonian diffeomorphisms that is generated by an invariant norm that satisfies the aforementioned continuity assumption, is either identically zero or equivalent to Hofer's metric.

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

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