Geometric variants of the Hofer norm

Details Geometric variants of the Hofer norm Geometric variants of the Hofer norm

2001-03-14

arxiv.org/abs/math/0103089v3

Journal of Symplectic Geometry, vol 1, 2002, 197-252

Mathematics

Symplectic Geometry

40 pages, v3: small correction to statement of Cor 1.19, two other minor corrections. v2: sign conventions changed, minor corr

Scientific paper

This note discusses some geometrically defined seminorms on the group $\Ham(M, \omega)$ of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M, \omega)$, 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 $\Ham(M, \omega)$ 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.

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