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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
##### Keywords:
Maslov index; 3-manifold; contact structure; tight
Full Text:
##### 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
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