no. 5
Anosov flows and non-Stein symplectic manifolds
Annales de l'Institut Fourier, Tome 45 (1995) no. 5, pp. 1407-1421.

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.

@article{AIF_1995__45_5_1407_0,
     author = {Mitsumatsu, Yoshihiko},
     title = {Anosov flows and {non-Stein} symplectic manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {1407--1421},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {5},
     year = {1995},
     doi = {10.5802/aif.1500},
     mrnumber = {96m:53029},
     zbl = {0834.53031},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1500/}
}
TY  - JOUR
AU  - Mitsumatsu, Yoshihiko
TI  - Anosov flows and non-Stein symplectic manifolds
JO  - Annales de l'Institut Fourier
PY  - 1995
SP  - 1407
EP  - 1421
VL  - 45
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.1500/
DO  - 10.5802/aif.1500
LA  - en
ID  - AIF_1995__45_5_1407_0
ER  - 
%0 Journal Article
%A Mitsumatsu, Yoshihiko
%T Anosov flows and non-Stein symplectic manifolds
%J Annales de l'Institut Fourier
%D 1995
%P 1407-1421
%V 45
%N 5
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.1500/
%R 10.5802/aif.1500
%G en
%F AIF_1995__45_5_1407_0
Mitsumatsu, Yoshihiko. Anosov flows and non-Stein symplectic manifolds. Annales de l'Institut Fourier, Tome 45 (1995) no. 5, pp. 1407-1421. doi : 10.5802/aif.1500. http://archive.numdam.org/articles/10.5802/aif.1500/

[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

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

[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 | Zbl

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

[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 | Zbl

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

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

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

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

[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

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

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

[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

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

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

Cité par Sources :