Hofer-Zehnder capacity and length minimizing paths in the Hofer norm

Abstract

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 $(M \times D(a))$ 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 $CP^2$ and in the blow-up of $CP^2$ 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.