Bi-invariant metrics on the group of symplectomorphisms
HTML articles powered by AMS MathViewer
- by Zhigang Han PDF
- Trans. Amer. Math. Soc. 361 (2009), 3343-3357 Request permission
Abstract:
This paper studies the extension of the Hofer metric and general Finsler metrics on the Hamiltonian symplectomorphism group $\textrm {Ham}(M,\omega )$ to the identity component $\textrm {Symp}_0(M,\omega )$ of the symplectomorphism group. In particular, we prove that the Hofer metric on $\textrm {Ham}(M,\omega )$ does not extend to a bi-invariant metric on $\textrm {Symp}_0(M,\omega )$ for many symplectic manifolds. We also show that for the torus $\mathbb T^{2n}$ with the standard symplectic form $\omega$, no Finsler metric on $\textrm {Ham}(\mathbb T^{2n},\omega )$ that satisfies a strong form of the invariance condition can extend to a bi-invariant metric on $\textrm {Symp}_0(\mathbb T^{2n},\omega )$. Another interesting result is that there exists no $C^1$-continuous bi-invariant metric on $\textrm {Symp}_0(\mathbb T^{2n},\omega )$.References
- Augustin Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv. 53 (1978), no. 2, 174–227 (French). MR 490874, DOI 10.1007/BF02566074
- Augustin Banyaga and Paul Donato, Lengths of contact isotopies and extensions of the Hofer metric, Ann. Global Anal. Geom. 30 (2006), no. 3, 299–312. MR 2256527, DOI 10.1007/s10455-005-9011-7
- Yakov Eliashberg and Leonid Polterovich, Bi-invariant metrics on the group of Hamiltonian diffeomorphisms, Internat. J. Math. 4 (1993), no. 5, 727–738. MR 1245350, DOI 10.1142/S0129167X93000352
- Michael Entov and Leonid Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 30 (2003), 1635–1676. MR 1979584, DOI 10.1155/S1073792803210011
- Z. Han, The bounded isometry conjecture for the Kodaira-Thurston manifold and the 4-torus, to appear in Israel J. Math., arXiv:0705.0762
- H. Hofer, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 1-2, 25–38. MR 1059642, DOI 10.1017/S0308210500024549
- François Lalonde and Dusa McDuff, The geometry of symplectic energy, Ann. of Math. (2) 141 (1995), no. 2, 349–371. MR 1324138, DOI 10.2307/2118524
- François Lalonde and Leonid Polterovich, Symplectic diffeomorphisms as isometries of Hofer’s norm, Topology 36 (1997), no. 3, 711–727. MR 1422431, DOI 10.1016/S0040-9383(96)00024-9
- François Lalonde and Charles Pestieau, Stabilisation of symplectic inequalities and applications, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 196, Amer. Math. Soc., Providence, RI, 1999, pp. 63–71. MR 1736214, DOI 10.1090/trans2/196/05
- Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. MR 1698616
- Yong-Geun Oh, Chain level Floer theory and Hofer’s geometry of the Hamiltonian diffeomorphism group, Asian J. Math. 6 (2002), no. 4, 579–624. MR 1958084, DOI 10.4310/AJM.2002.v6.n4.a1
- Yaron Ostrover and Roy Wagner, On the extremality of Hofer’s metric on the group of Hamiltonian diffeomorphisms, Int. Math. Res. Not. 35 (2005), 2123–2141. MR 2181789, DOI 10.1155/IMRN.2005.2123
- Leonid Polterovich, Symplectic displacement energy for Lagrangian submanifolds, Ergodic Theory Dynam. Systems 13 (1993), no. 2, 357–367. MR 1235478, DOI 10.1017/S0143385700007410
- Leonid Polterovich, The geometry of the group of symplectic diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001. MR 1826128, DOI 10.1007/978-3-0348-8299-6
- Leonid Polterovich, Hofer’s diameter and Lagrangian intersections, Internat. Math. Res. Notices 4 (1998), 217–223. MR 1609620, DOI 10.1155/S1073792898000178
- Matthias Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193 (2000), no. 2, 419–461. MR 1755825, DOI 10.2140/pjm.2000.193.419
- P. Seidel, $\pi _1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997), no. 6, 1046–1095. MR 1487754, DOI 10.1007/s000390050037
Additional Information
- Zhigang Han
- Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Amherst, Massachusetts 01003-9305
- Email: han@math.umass.edu
- Received by editor(s): October 1, 2007
- Published electronically: December 31, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 3343-3357
- MSC (2000): Primary 53D35; Secondary 57R17
- DOI: https://doi.org/10.1090/S0002-9947-08-04713-2
- MathSciNet review: 2485430