1999/05/18 by Jennifer Slimowitz, Slimowitz, Jennifer
Chemistry · Mathematics · #Advanced Operator Algebra Research #Geometric and Algebraic Topology #Supramolecular Chemistry and Complexes #math.DG #math.SG #msc:53C15 #msc:58B20 #msc:58D05 #msc:58F05
paper · pdf · doi:10.48550/arxiv.math/9905105
34 pages, LaTeX2e, 9 figures. Submitted to Transactions of the AMS
arxiv created 1999/05/18 · arxiv updated 2009/11/30
We use the criteria of Lalonde and McDuff to determine a new class of examples of length minimizing paths in the group Ham(M). For a compact symplectic manifold M of dimension two or four, we show that a path in Ham(M), generated by an autonomous Hamiltonian and starting at the identity, which induces no non-constant closed trajectories of points in M, is length minimizing among homotopic paths. The major step in the proof involves determining an upper bound for the Hofer-Zehnder capacity for symplectic manifolds of the type (M × D(a)) where M is compact and has dimension two or four. In the appendix, we give an alternate proof of Polterovich's result that rotation in CP2 and in the blow-up of CP2 at one point is a length minimizing path with respect to the Hofer norm. Here we use the Gromov capacity and describe the necessary ball embeddings.