Geometric Invariants of the Hofer Norm

This note discusses some geometrically defined seminorms on the group Ham (M,ω) of Hamiltonian diffeomorphisms of a closed symplectic manifold (M,ω), giving conditions under which they are nondegenerate and explaining their relation to the Hofer norm. As a consequence we show that if an element in Ham (M,ω) is sufficiently close to identity in the C²-topology then it may be joined to the identity by a path whose Hofer length is minimal among all paths, not just among paths in the same homotopy class relative to endpoints. Thus, true geodesics always exist for the Hofer norm. The main step in the proof is to show that a "weighted" version of the nonsqueezing theorem holds for all fibrations over S² generated by sufficiently short loops. Further, an example is given showing that the Hofer norm may differ from the sum of one sided seminorms.

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Published 1 January 2001