m at h . SG ] 4 O ct 2 00 6 Geometric variants of the Hofer norm

This note discusses some geometrically defined seminorms on the group Ham(M, ω) of Hamilto-nian 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 the identity in the C 2-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 S 2 generated by sufficiently short loops. Further, an example is given showing that the Hofer norm may differ from the sum of the one sided seminorms. The revision of August 2006 corrects minor mistakes in the statements of Corollary 1.19 and Lemma 4.8.