Anosov flows and non-Stein symplectic manifolds
Mitsumatsu, Yoshihiko
Annales de l'Institut Fourier, Tome 45 (1995), p. 1407-1421 / Harvested from Numdam
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La construction par McDuff de variétés symplectiques de dimension 4 à bord non connexe de type contact, est simplifiée et généralisée en termes du produit scalaire d’enlacement sur le dual des algèbres de Lie de dimension 3. Cela nous amène à observer que les flots d’Anosov donnent des structures de bi-contact, c’est-à-dire une paire transversale de structures de contact avec orientations opposées. De plus, on voit que la construction se généralise aux variétés de dimension 3 qui admettent un flot d’Anosov avec un volume invariant lisse. Enfin, de nouveaux exemples de structure de bi-contact sont donnés et les problèmes dynamiques autour des structures de bi-contact sont proposés.

We simplify and generalize McDuff’s construction of symplectic 4-manifolds with disconnected boundary of contact type in terms of the linking pairing defined on the dual of 3-dimensional Lie algebras. This leads us to an observation that an Anosov flow gives rise to a bi-contact structure, i.e. a transverse pair of contact structures with different orientations, and the construction turns out to work for 3-manifolds which admit Anosov flows with smooth invariant volume. Finally, new examples of bi-contact structures are given and related dynamical problems around bi-contact structures are raised.

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     author = {Mitsumatsu, Yoshihiko},
     title = {Anosov flows and non-Stein symplectic manifolds},
     journal = {Annales de l'Institut Fourier},
     volume = {45},
     year = {1995},
     pages = {1407-1421},
     doi = {10.5802/aif.1500},
     mrnumber = {96m:53029},
     zbl = {0834.53031},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1995__45_5_1407_0}
}

Mitsumatsu, Yoshihiko. Anosov flows and non-Stein symplectic manifolds. Annales de l'Institut Fourier, Tome 45 (1995) pp. 1407-1421. doi : 10.5802/aif.1500. http://gdmltest.u-ga.fr/item/AIF_1995__45_5_1407_0/

[1] D. Bennequin, Topologie symplectique, convexité holomorphe et structures de contact, d'après Y. Eliashberg, D. McDuff et al., Séminaire Bourbaki, 725 (1989-1990). | Numdam | Zbl 0755.32009

[2] Y. Eliashberg, Topological characterization of Stein Manifolds of dimension > 2, International J. Math., 1 (1990), 19-46. | Zbl 0699.58002

[3] Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier, Grenoble, 42 (1-2) (1991), 165-192. | Numdam | MR 93k:57029 | Zbl 0756.53017

[4] Y. Eliashberg and M. Gromov, Convex symplectic manifolds, Proc. Symp. Pure Math. A.M.S., 52 (2) (1991), 135-162. | MR 93f:58073 | Zbl 0742.53010

[5] P. Foulon, Preprint in preparation.

[6] H. Geiges, Preprints.

[7] E. Ghys, Flots d'Anosov dont les feuilletages stables sont différentiables, Ann. Scient. École Norm. Sup., 20 (1987), 251-270. | Numdam | MR 89h:58153 | Zbl 0663.58025

[8] S. Goodman, Dehn surgery and Anosov flows in Proc. Geom. Dynamics Conf., Springer Lecture Notes in Mathematics, 1007 (1983). | Zbl 0532.58021

[9] R.C. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall Inc., Englewood Cliffs N.J., 1965. | MR 31 #4927 | Zbl 0141.08601

[10] M. Handel and W.P. Thurston, Anosov flows on new 3-manifolds, Invent. Math., 59 (1980), 95-103. | MR 81i:58032 | Zbl 0435.58019

[11] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Springer Lecture Notes in Mathematics, 583 (1977). | MR 58 #18595 | Zbl 0355.58009

[12] F. Laudenbach, Orbites périodiques et courbes pseudo-holomorphes, application à la conjecture de Weinstein en dimension 3, d'après H. Hofer et al., Séminaire Bourbaki, 786 (1993-1994). | Numdam | Zbl 0853.57013

[13] D. Mcduff, Examples of simply-connected non-Kählerian manifolds, J. Diff. Geom., 20 (1984), 267-277. | MR 86c:57036 | Zbl 0567.53031

[14] D. Mcduff, Symplectic manifolds with contact type boundaries, Invent. Math., 103 (1991), 651-671. | MR 92e:53042 | Zbl 0719.53015

[15] Th. Peternell, Pseudoconvexity, the Levi problem and vanishing theorems, Encyclopeadia of Mathematical Sciences, 74, Several complex variables VII, Chapter VIII, Springer-Verlag, Berlin (1994). | Zbl 0811.32011

[16] W.P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc., 55 (1976), 467-468. | MR 53 #6578 | Zbl 0324.53031

[17] A. Weinstein, On the hypotheses of Rabinowtz's periodic orbit theorems, J. Diff. Eq., 33 (1979), 353-358. | MR 81a:58030b | Zbl 0388.58020