×

zbMATH — the first resource for mathematics

A characterisation of the tight three-sphere. (English) Zbl 0861.57026
Duke Math. J. 81, No. 1, 159-226 (1995); correction ibid. 89, No. 3, 603-617 (1997).
Let \(M\) be a compact oriented 3-manifold and assume that \(M\) carries a positive co-orientable contact structure with plane distribution \(\xi\subset TM\). The contact structure \(\xi\) is said to be tight if there is no disk \(F\) imbedded in \(M\) such that \(T_z(\partial F)\subset\xi_z\) and \(T_zF\not\subset\xi_z\) for all \(z\in\partial F\). The authors’ main result gives a characterization of the positive tight contact structure on \(S^3\):
\(M\) as above is contact isomorphic to \(S^3\) with its positive tight contact structure iff there exists a contact form \(\lambda\) defining \(\xi\) with the following property. The Reeb vector field \(X\) associated to \(\lambda\) admits a nondegenerate periodic orbit \(P_0\) spanning an imbedded disk \(F\) whose interior is traversal to \(X\) and whose index is \(\mu(P_0,F)=3\).
Reviewer: G.Tóth (Camden)

MSC:
57N12 Topology of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] C. Abbas and H. Hofer, Holomorphic Curves and Global Questions in Contact Geometry , Birkhäuser, Boston, to be published.
[2] D. Bennequin, Entrelacements et équations de Pfaff , Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982), Astérisque, vol. 107, Soc. Math. France, Paris, 1983, pp. 87-161. · Zbl 0573.58022
[3] E. Bishop, Differentiable manifolds in complex Euclidean space , Duke Math. J. 32 (1965), 1-21. · Zbl 0154.08501 · doi:10.1215/S0012-7094-65-03201-1
[4] J. Cerf, Sur les difféomorphismes de la sphère de dimension trois \((\Gamma _4=0)\) , Lecture Notes in Mathematics, No. 53, Springer-Verlag, Berlin, 1968. · Zbl 0164.24502 · doi:10.1007/BFb0060395
[5] C. Conley and E. Zehnder, Morse type index theory for flows and periodic solutions for Hamiltonian equations , Comm. Pure Appl. Math. 37 (1984), no. 2, 207-253. · Zbl 0559.58019 · doi:10.1002/cpa.3160370204
[6] Y. Eliashberg, Classification of contact structures on \(\mathbf R^ 3\) , Internat. Math. Res. Notices (1993), no. 3, 87-91. · Zbl 0784.53022 · doi:10.1155/S107379289300008X
[7] Y. Eliashberg, Classification of overtwisted contact structures on three manifolds , Invent. Math. 98 (1989), no. 3, 623-637. · Zbl 0684.57012 · doi:10.1007/BF01393840 · eudml:143746
[8] Y. Eliashberg, Contact \(3\)-manifolds twenty years since J. Martinet’s work , Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 165-192. · Zbl 0756.53017 · doi:10.5802/aif.1288 · numdam:AIF_1992__42_1-2_165_0 · eudml:74949
[9] Y. Eliashberg, Filling by holomorphic discs and its applications , Geometry of Low-dimensional Manifolds, 2 (Durham, 1989), London Math. Soc. Lecture Note Ser., vol. 151, Cambridge Univ. Press, Cambridge, 1990, pp. 45-67. · Zbl 0731.53036
[10] Y. Eliashberg, Legendrian and transversal knots in tight contact \(3\)-manifolds , Topological Methods in Modern Mathematics (Stony Brook, NY, 1991) eds. L. Goldberg and A. Phillips, Publish or Perish, Houston, Tex., 1993, pp. 171-193. · Zbl 0809.53033
[11] Y. Eliashberg and H. Hofer, A Hamiltonian characterization of the three-ball , Differential Integral Equations 7 (1994), no. 5-6, 1303-1324. · Zbl 0803.58045
[12] A. Floer, An instanton-invariant for \(3\)-manifolds , Comm. Math. Phys. 118 (1988), no. 2, 215-240. · Zbl 0684.53027 · doi:10.1007/BF01218578
[13] A. Floer, Morse theory for Lagrangian intersections , J. Differential Geom. 28 (1988), no. 3, 513-547. · Zbl 0674.57027
[14] A. Floer, Symplectic fixed points and holomorphic spheres , Comm. Math. Phys. 120 (1989), no. 4, 575-611. · Zbl 0755.58022 · doi:10.1007/BF01260388
[15] A. Floer, The unregularized gradient flow of the symplectic action , Comm. Pure Appl. Math. 41 (1988), no. 6, 775-813. · Zbl 0633.53058 · doi:10.1002/cpa.3160410603
[16] A. Floer, H. Hofer, and D. Salamon, Transversality in elliptic Morse theory for the symplectic action , Duke Math. J. 80 (1995), no. 1, 251-292. · Zbl 0846.58025 · doi:10.1215/S0012-7094-95-08010-7
[17] J. Franks, Geodesics on \(S^2\) and periodic points of annulus homeomorphisms , Invent. Math. 108 (1992), no. 2, 403-418. · Zbl 0766.53037 · doi:10.1007/BF02100612 · eudml:144001
[18] E. Giroux, Convexité en topologie de contact , Comment. Math. Helv. 66 (1991), no. 4, 637-677. · Zbl 0766.53028 · doi:10.1007/BF02566670 · eudml:140253
[19] M. Gromov, Pseudoholomorphic curves in symplectic manifolds , Invent. Math. 82 (1985), no. 2, 307-347. · Zbl 0592.53025 · doi:10.1007/BF01388806 · eudml:143289
[20] H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three , Invent. Math. 114 (1993), no. 3, 515-563. · Zbl 0797.58023 · doi:10.1007/BF01232679 · eudml:144157
[21] H. Hofer and D. Salamon, Floer homology and Novikov rings , The Floer Memorial Volume, Progr. Math., vol. 133, Birkhäuser, Basel, 1995, pp. 483-524. · Zbl 0842.58029
[22] H. Hofer and C. Viterbo, The Weinstein conjecture in the presence of holomorphic spheres , Comm. Pure Appl. Math. 45 (1992), no. 5, 583-622. · Zbl 0773.58021 · doi:10.1002/cpa.3160450504
[23] H. Hofer, K. Wysocki, and E. Zehnder, The dynamics on a strictly convex energy surface in \(R^4\) , · Zbl 0944.37031
[24] H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectisations I: Asymptotics , to appear in Analyse Nonlinéaire, May 1996. · Zbl 0861.58018 · numdam:AIHPC_1996__13_3_337_0 · numdam:AIHPC_1998__15_4_535_0 · eudml:78447
[25] H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectisations II: Embedding controls and algebraic invariants , to appear in Geom. Funct. Anal. 5, 1995. · Zbl 0845.57027 · doi:10.1007/BF01895669 · eudml:58191
[26] H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectisations III: Fredholm theory , · Zbl 0924.58003
[27] H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics , Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. · Zbl 0805.58003
[28] J. Martinet, Formes de contact sur les variétés de dimension \(3\) , Proceedings of Liverpool Singularities Symposium, II (1969/1970), Springer, Berlin, 1971, 142-163. Lecture Notes in Math., Vol. 209. · Zbl 0215.23003
[29] D. McDuff, The local behaviour of holomorphic curves in almost complex \(4\)-manifolds , J. Differential Geom. 34 (1991), no. 1, 143-164. · Zbl 0736.53038
[30] D. McDuff and D. Salamon, Introduction to Symplectic Topology , Oxford University Press, Oxford, to be published. · Zbl 0844.58029
[31] D. McDuff and D. Salamon, \(J\)-Holomorphic Curves and Quantum Cohomology , University Lecture Series, vol. 6, Amer. Math. Soc., Providence, 1994. · Zbl 0809.53002
[32] M. Micallef and B. White, The structure of branch points in minimal surfaces and in pseudoholomorphic curves , Ann. of Math. (2) 141 (1995), no. 1, 35-85. JSTOR: · Zbl 0873.53038 · doi:10.2307/2118627 · links.jstor.org
[33] Y. G. Oh, Removal of boundary singularities of pseudo-holomorphic curves with Lagrangian boundary conditions , Comm. Pure Appl. Math. 45 (1992), no. 1, 121-139. · Zbl 0743.58018 · doi:10.1002/cpa.3160450106
[34] T. H. Parker and J. G. Wolfson, Pseudo-holomorphic maps and bubble trees , preprint, 1991. · Zbl 0759.53023 · doi:10.1007/BF02921330
[35] J. Robbin and D. Salamon, The spectral flow and the Maslov index , to appear in J. London Math. Soc. · Zbl 0859.58025 · doi:10.1112/blms/27.1.1
[36] D. Rohlfson, Knots , Publish or Perish, Houston, Tex., 1976.
[37] S. Smale, Diffeomorphisms of the \(2\)-sphere , Proc. Amer. Math. Soc. 10 (1959), 621-626. JSTOR: · Zbl 0118.39103 · doi:10.2307/2033664 · links.jstor.org
[38] R. Ye, Filling by holomorphic disks in symplectic \(4\)-manifolds , · Zbl 0936.53047
[39] R. Ye, Gromov’s compactness theorem for pseudo holomorphic curves , Trans. Amer. Math. Soc. 342 (1994), no. 2, 671-694. JSTOR: · Zbl 0810.53024 · doi:10.2307/2154647 · links.jstor.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.