## Geometry & Topology

### Hofer–Zehnder capacity and length minimizing Hamiltonian paths

#### Abstract

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 $S1$–invariant virtual moduli cycles. As a corollary, we find that any semifree action of $S1$ 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.

#### Article information

Source
Geom. Topol., Volume 5, Number 2 (2001), 799-830.

Dates
Received: 12 January 2001
Revised: 9 October 2001
Accepted: 9 November 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883044

Digital Object Identifier
doi:10.2140/gt.2001.5.799

Mathematical Reviews number (MathSciNet)
MR1871405

Zentralblatt MATH identifier
1002.57056

#### Citation

McDuff, Dusa; Slimowitz, Jennifer. Hofer–Zehnder capacity and length minimizing Hamiltonian paths. Geom. Topol. 5 (2001), no. 2, 799--830. doi:10.2140/gt.2001.5.799. https://projecteuclid.org/euclid.gt/1513883044

#### References

• M Abreu, D McDuff, Topology of symplectomorphism groups of rational ruled surfaces, Journal of the Amer. Math. Soc. 13 (2000) 971–1009
• M Bialy, L Polterovich, Geodesics of Hofer's metric on the group of Hamiltonian diffeomorphisms, Duke J. Math. 76 (1994) 273–292
• M Entov, $K$–area, Hofer metric and geometry of conjugacy classes in Lie groups, Geometric and Functional Analysis (2001)
• A Floer, Symplectic fixed points and holomorphic spheres, Communications in Mathematical Physics, 120 (1989) 575–611
• A Floer, H Hofer, D Salamon, Transversality in Elliptic Morse Theory for the Symplectic Action, Duke Math. J. 80 (1995) 251–292
• M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347
• H. Hofer, Estimates for the energy of a symplectic map, Commentarii Mathematici Helvetici, 68 (1993) 48–72
• H Hofer, C Viterbo, The Weinstein Conjecture in the Presence of Holomorphic Spheres, Comm. on Pure and Applied Math. XLV (1992) 583–622
• H Hofer, E Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhauser, Boston, MA (1994)
• F Lalonde, D McDuff, Hofer's $L^{\infty}$–geometry: energy and stability of Hamiltonian flows, parts I and II, Invent. Math. 122 (1995) 1–33 and 35–69
• Gang Liu, Gang Tian, Weinstein Conjecture and GW Invariants, Commun. Contemp. Math. 2 (2000) 405–459
• Gang Liu, Gang Tian, Floer homology and Arnold conjecture, Journ. Diff. Geom. 49 (1998) 1–74
• Gang Liu, Gang Tian, On the equivalence of multiplicative structures in Floer Homology and Quantum Homology, Acta Math. Sinica, 15 (1999)
• GuangCun Lu, The Weinstein conjecture on some symplectic manifolds containing the holomorphic spheres, Kyushu J. Math. 52 (1998) 331–51 and 54 (2000) 181–2
• D McDuff, The virtual moduli cycle, Amer. Math. Soc. Transl. (2) 196 (1999) 73–102
• D McDuff, Quantum homology of fibrations over $S^2$, Internat. Math. Journal, 11 (2000) 665–721
• D McDuff, D Salamon, Introduction to Symplectic Topology, 2nd edition, Oxford University Press, Oxford, England (1998)
• D McDuff, D Salamon, J-Holomorphic Curves and Quantum Cohomology, University Lecture Series 6, American Mathematical Society (1994).
• D McDuff, S Tolman, Topological properties of Hamiltonian circle actions, in preparation December 2001
• J Moser, Addendum to “Periodic Orbits near Equilibrium and a theorem by Alan Weinstein”, Comm. Pure and Appl. Math. 31 (1978) 529–530
• S Piunikhin, D Salamon, M Schwarz, Symplectic Floer–Donaldson theory and Quantum Cohomology, from: “Contact and Symplectic Geometry”, (C Thomas, editor), Proceedings of the 1994 Newton Institute Conference, CUP, Cambridge (1996)
• L Polterovich, Gromov's K–area and symplectic rigidity, Geometric and Functional Analysis, 6 (1996) 726–39
• L Polterovich, Hamiltonian loops and Arnold's principle, Amer. Math. Soc. Transl. (2) 180 (1997) 181-187
• L Polterovich, Symplectic aspects of the first eigenvalue, Journ. fur die Riene und angew. Math. 502 (1998) 1–17
• L Polterovich, The Geometry of the group of symplectomorphisms, Birkhäuser (2001)
• D Salamon, E Zehnder, Morse theory for Periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure and Appl. Math. 45 (1992) 1303–1360
• M Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pac. Journ. Math. 193 (2000) 419–461
• C Siegel, J Moser, Lectures on Celestial Mechanics, Springer Verlag (1971)
• K Siburg, New minimal geodesics in the group of symplectic diffeomorphisms, Calc. Var 3 (1995) 299–309.
• J Slimowitz, PhD thesis, Stony Brook (1998)
• I Ustilovsky, Conjugate points on geodesics of Hofer's metric, Diff. Geometry and its Appl. 6 (1994) 327–342
• A Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math. 20 (1973) 47–57