Many closed symplectic manifolds have infinite Hofer–Zehnder capacity
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Abstract:
We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus $T^{2n}$ ($n\geq 2$) with an irrational symplectic structure). The underlying smooth manifolds of our examples include, for instance: the $K3$ surface and also infinitely many smooth manifolds homeomorphic but not diffeomorphic to it; infinitely many minimal four-manifolds having any given finitely-presented group as their fundamental group; and simply connected minimal four-manifolds realizing all but finitely many points in the first quadrant of the geography plane below the line corresponding to signature $3$. The examples are constructed by performing symplectic sums along suitable tori and then perturbing the symplectic form in such a way that hypersurfaces near the “neck” in the symplectic sum have no closed characteristics. We conjecture that any closed symplectic four-manifold with $b^+>1$ admits symplectic forms with a similar property.References
- Anar Akhmedov, Scott Baldridge, R. İnanç Baykur, Paul Kirk, and B. Doug Park, Simply connected minimal symplectic 4-manifolds with signature less than $-1$, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 1, 133–161. MR 2578606, DOI 10.4171/JEMS/192
- Ronald Fintushel and Ronald J. Stern, Rational blowdowns of smooth $4$-manifolds, J. Differential Geom. 46 (1997), no. 2, 181–235. MR 1484044
- Ronald Fintushel and Ronald J. Stern, Knots, links, and $4$-manifolds, Invent. Math. 134 (1998), no. 2, 363–400. MR 1650308, DOI 10.1007/s002220050268
- Viktor L. Ginzburg, An embedding $S^{2n-1}\to \textbf {R}^{2n}$, $2n-1\geq 7$, whose Hamiltonian flow has no periodic trajectories, Internat. Math. Res. Notices 2 (1995), 83–97. MR 1317645, DOI 10.1155/S1073792895000079
- Viktor L. Ginzburg, A smooth counterexample to the Hamiltonian Seifert conjecture in $\mathbf R^6$, Internat. Math. Res. Notices 13 (1997), 641–650. MR 1459629, DOI 10.1155/S1073792897000421
- Viktor L. Ginzburg and Başak Z. Gürel, A $C^2$-smooth counterexample to the Hamiltonian Seifert conjecture in $\Bbb R^4$, Ann. of Math. (2) 158 (2003), no. 3, 953–976. MR 2031857, DOI 10.4007/annals.2003.158.953
- Robert E. Gompf, Some new symplectic $4$-manifolds, Turkish J. Math. 18 (1994), no. 1, 7–15. MR 1270434
- Robert E. Gompf, A new construction of symplectic manifolds, Ann. of Math. (2) 142 (1995), no. 3, 527–595. MR 1356781, DOI 10.2307/2118554
- Robert E. Gompf and András I. Stipsicz, $4$-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. MR 1707327, DOI 10.1090/gsm/020
- Mark J. Gotay, On coisotropic imbeddings of presymplectic manifolds, Proc. Amer. Math. Soc. 84 (1982), no. 1, 111–114. MR 633290, DOI 10.1090/S0002-9939-1982-0633290-X
- H. Hofer and E. Zehnder, Periodic solutions on hypersurfaces and a result by C. Viterbo, Invent. Math. 90 (1987), no. 1, 1–9. MR 906578, DOI 10.1007/BF01389030
- Helmut Hofer and Eduard Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. MR 1306732, DOI 10.1007/978-3-0348-8540-9
- Eleny-Nicoleta Ionel and Thomas H. Parker, The Gromov invariants of Ruan-Tian and Taubes, Math. Res. Lett. 4 (1997), no. 4, 521–532. MR 1470424, DOI 10.4310/MRL.1997.v4.n4.a9
- Eleny-Nicoleta Ionel and Thomas H. Parker, Gromov invariants and symplectic maps, Math. Ann. 314 (1999), no. 1, 127–158. MR 1689266, DOI 10.1007/s002080050289
- Eleny-Nicoleta Ionel and Thomas H. Parker, The symplectic sum formula for Gromov-Witten invariants, Ann. of Math. (2) 159 (2004), no. 3, 935–1025. MR 2113018, DOI 10.4007/annals.2004.159.935
- Eugene Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), no. 3, 247–258. MR 1338784, DOI 10.4310/MRL.1995.v2.n3.a2
- Tian-Jun Li and Ai-Ko Liu, The equivalence between $\textrm {SW}$ and $\textrm {Gr}$ in the case where $b^+=1$, Internat. Math. Res. Notices 7 (1999), 335–345. MR 1683312, DOI 10.1155/S1073792899000173
- Tian-Jun Li and Ai-Ko Liu, Uniqueness of symplectic canonical class, surface cone and symplectic cone of 4-manifolds with $B^+=1$, J. Differential Geom. 58 (2001), no. 2, 331–370. MR 1913946
- Ai-Ko Liu, Some new applications of general wall crossing formula, Gompf’s conjecture and its applications, Math. Res. Lett. 3 (1996), no. 5, 569–585. MR 1418572, DOI 10.4310/MRL.1996.v3.n5.a1
- Junho Lee and Thomas H. Parker, A structure theorem for the Gromov-Witten invariants of Kähler surfaces, J. Differential Geom. 77 (2007), no. 3, 483–513. MR 2362322
- Guangcun Lu, Gromov-Witten invariants and pseudo symplectic capacities, Israel J. Math. 156 (2006), 1–63. MR 2282367, DOI 10.1007/BF02773823
- Charles-Michel Marle, Sous-variétés de rang constant d’une variété symplectique, Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982) Astérisque, vol. 107, Soc. Math. France, Paris, 1983, pp. 69–86 (French). MR 753130
- Dusa McDuff, Immersed spheres in symplectic $4$-manifolds, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 369–392 (English, with French summary). MR 1162567, DOI 10.5802/aif.1296
- Dusa McDuff, From symplectic deformation to isotopy, Topics in symplectic $4$-manifolds (Irvine, CA, 1996) First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA, 1998, pp. 85–99. MR 1635697
- Dusa McDuff and Margaret Symington, Associativity properties of the symplectic sum, Math. Res. Lett. 3 (1996), no. 5, 591–608. MR 1418574, DOI 10.4310/MRL.1996.v3.n5.a3
- John D. McCarthy and Jon G. Wolfson, Symplectic normal connect sum, Topology 33 (1994), no. 4, 729–764. MR 1293308, DOI 10.1016/0040-9383(94)90006-X
- Margaret Symington, Symplectic rational blowdowns, J. Differential Geom. 50 (1998), no. 3, 505–518. MR 1690738
- Clifford Henry Taubes, Seiberg Witten and Gromov invariants for symplectic $4$-manifolds, First International Press Lecture Series, vol. 2, International Press, Somerville, MA, 2000. Edited by Richard Wentworth. MR 1798809
- W. P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), no. 2, 467–468. MR 402764, DOI 10.1090/S0002-9939-1976-0402764-6
- Michael Usher, Minimality and symplectic sums, Int. Math. Res. Not. , posted on (2006), Art. ID 49857, 17. MR 2250015, DOI 10.1155/IMRN/2006/49857
- E. Zehnder, Remarks on periodic solutions on hypersurfaces, Periodic solutions of Hamiltonian systems and related topics (Il Ciocco, 1986) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 209, Reidel, Dordrecht, 1987, pp. 267–279. MR 920629
Additional Information
- Michael Usher
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- Email: usher@math.uga.edu
- Received by editor(s): January 31, 2011
- Published electronically: May 18, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 5913-5943
- MSC (2010): Primary 53D35, 37J45
- DOI: https://doi.org/10.1090/S0002-9947-2012-05623-6
- MathSciNet review: 2946937