## Duke Mathematical Journal

### A characterisation of the tight three-sphere

#### Article information

Source
Duke Math. J., Volume 81, Number 1 (1995), 159-226.

Dates
First available in Project Euclid: 19 February 2004

https://projecteuclid.org/euclid.dmj/1077245466

Digital Object Identifier
doi:10.1215/S0012-7094-95-08111-3

Mathematical Reviews number (MathSciNet)
MR1381975

Zentralblatt MATH identifier
0861.57026

#### Citation

Hofer, H.; Wysocki, K.; Zehnder, E. A characterisation of the tight three-sphere. Duke Math. J. 81 (1995), no. 1, 159--226. doi:10.1215/S0012-7094-95-08111-3. https://projecteuclid.org/euclid.dmj/1077245466

#### 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.
• [3] E. Bishop, Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1–21.
• [4] J. Cerf, Sur les difféomorphismes de la sphère de dimension trois $(\Gamma \sb{4}=0)$, Lecture Notes in Mathematics, No. 53, Springer-Verlag, Berlin, 1968.
• [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.
• [6] Y. Eliashberg, Classification of contact structures on $\bold R\sp 3$, Internat. Math. Res. Notices (1993), no. 3, 87–91.
• [7] Y. Eliashberg, Classification of overtwisted contact structures on three manifolds, Invent. Math. 98 (1989), no. 3, 623–637.
• [8] Y. Eliashberg, Contact $3$-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 165–192.
• [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.
• [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.
• [11] Y. Eliashberg and H. Hofer, A Hamiltonian characterization of the three-ball, Differential Integral Equations 7 (1994), no. 5-6, 1303–1324.
• [12] A. Floer, An instanton-invariant for $3$-manifolds, Comm. Math. Phys. 118 (1988), no. 2, 215–240.
• [13] A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547.
• [14] A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), no. 4, 575–611.
• [15] A. Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), no. 6, 775–813.
• [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.
• [17] J. Franks, Geodesics on $S^{2}$ and periodic points of annulus homeomorphisms, Invent. Math. 108 (1992), no. 2, 403–418.
• [18] E. Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991), no. 4, 637–677.
• [19] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347.
• [20] H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993), no. 3, 515–563.
• [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.
• [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.
• [23] H. Hofer, K. Wysocki, and E. Zehnder, The dynamics on a strictly convex energy surface in $R^{4}$, preprint.
• [24] H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectisations I: Asymptotics, to appear in Analyse Nonlinéaire, May 1996.
• [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.
• [26] H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectisations III: Fredholm theory, preprint.
• [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.
• [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.
• [29] D. McDuff, The local behaviour of holomorphic curves in almost complex $4$-manifolds, J. Differential Geom. 34 (1991), no. 1, 143–164.
• [30] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford University Press, Oxford, to be published.
• [31] D. McDuff and D. Salamon, $J$-Holomorphic Curves and Quantum Cohomology, University Lecture Series, vol. 6, Amer. Math. Soc., Providence, 1994.
• [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.
• [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.
• [34] T. H. Parker and J. G. Wolfson, Pseudo-holomorphic maps and bubble trees, preprint, 1991.
• [35] J. Robbin and D. Salamon, The spectral flow and the Maslov index, to appear in J. London Math. Soc.
• [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.
• [38] R. Ye, Filling by holomorphic disks in symplectic $4$-manifolds, preprint.
• [39] R. Ye, Gromov's compactness theorem for pseudo holomorphic curves, Trans. Amer. Math. Soc. 342 (1994), no. 2, 671–694.