Celebratio Mathematica — McDuff

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Works connected to Jennifer Robin Slimowitz

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D. McDuff and J. Slimowitz: “Hofer–Zehnder capacity and length minimizing Hamiltonian paths,” Geom. Topol. 5 : 2 (2001), pp. 799–830. MR 1871405 Zbl 1002.57056 article

We use the criteria of Lalonde and McDuff to show that a path that is generated by a generic autonomous Hamiltonian is length minimizing with respect to the Hofer norm among all homotopic paths provided that it induces no non-constant closed trajectories in \( M \). This generalizes a result of Hofer for symplectomorphisms of Euclidean space. The proof for general \( M \) uses Liu–Tian’s construction of \( S^1 \)-invariant virtual moduli cycles. As a corollary, we find that any semifree action of \( S^1 \) on \( M \) gives rise to a nontrivial element in the fundamental group of the symplectomorphism group of \( M \). We also establish a version of the area-capacity inequality for quasicylinders.