2011/07/31 by Michael Usher
Mathematics · #math.SG
published as Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 1, 57--128 · 62 pages. v2: added details concerning a key computation of the boundary depths of certain Hamiltonians, including a new appendix about transversality for t-independent Floer trajectories. v3: minor corrections. To appear in Ann. Sci. Ec. Norm. Sup
arxiv created 2012/11/01 · arxiv updated 2014/09/10
We show that if (M,ω) is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer's metric on the group of Hamiltonian diffeomorphisms of (M,ω) has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer's metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in M x M when M satisfies the above dynamical condition. To prove this, we use the properties of a Floer-theoretic quantity called the boundary depth, which measures the nontriviality of the boundary operator on the Floer complex in a way that encodes robust symplectic-topological information.