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×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
C
P
2
and in the blow-up of
C
P
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.