Hofer–Zehnder capacity and length minimizing Hamiltonian paths

By D. Mcduff, J. Slimowitz. 2001

62 citations recorded by the source corpus.

We use the criteria of Lalonde and McDu 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 nd 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 .W e also establish a version of the area-capacity inequality for quasicylinders.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/94bf93ea72bc2420a519800f75bce44693cced64

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Citations in the corpus

LECTURES ON FLOER THEORY AND SPECTRAL INVARIANTS OF HAMILTONIAN FLOWS (2006, background): “The following conjecture was raised by Polterovich, Conjecture 12.6.D [Po2]. (See also [Po3], [LM2] and [ MSl ].)”
Chain level Floer theory and Hofer's geometry of the Hamiltonian diffeomorphism group (2001, methodology): “The “slowness” condition introduced in [En], [ MS ] is related to this problem.”
Topological properties of Hamiltonian circle actions (2004, result): “This generalizes the result of McDuff–Slimowitz [21] stating that ΛK is essential if the action is semifree.”
THE SHARP ENERGY-CAPACITY INEQUALITY (2008, methodology): “To state this result, we introduce some terminology, which is borrowed from [25] (readers familiar with [16] should note that our definition of “slow” is different from the one there).”
Quantitative symplectic geometry (2005, unknown): “Since then, lots of new capacities have been defined [16, 30, 32, 44, 49, 59, 60, 90, 99] and they were further studied in [1, 2, 8, 9, 17, 26, 21, 28, 31, 35, 37, 38, 41, 42, 43, 46, 48, 50, 52, 56, 57, 58, 61, 62, 63, 64, 65, 66, 68, 74, 75, 76, 88, 89, 91, 92, 94, 97, 98].”
Gromov-Witten invariants and pseudo symplectic capacities (2001, methodology): “pendix we discuss some related results on the Gromov-Witten invariants of product manifolds. Acknowledgements. I am deeply grateful to Professor Dusa McDuﬀ for kindly explaining to me some details of [McSl], sending me a preliminary version of her preprint [Mc3], correcting many errors, giving many ”
The Gromov width of 4-dimensional tori (2011, background): “Computations and estimates of the Gromov width for various examples can be found in [4, 5, 6, 15, 18, 19, 21, 22, 29, 34, 38, 41, 42].”