Anosov flows and non-Stein symplectic manifolds

Yoshihiko Mitsumatsu

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 5, page 1407-1421
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Mitsumatsu, Yoshihiko. "Anosov flows and non-Stein symplectic manifolds." Annales de l'institut Fourier 45.5 (1995): 1407-1421. <http://eudml.org/doc/75164>.

@article{Mitsumatsu1995,
abstract = {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.},
author = {Mitsumatsu, Yoshihiko},
journal = {Annales de l'institut Fourier},
keywords = {Anosov flows; contact structures; convex symplectic structures},
language = {eng},
number = {5},
pages = {1407-1421},
publisher = {Association des Annales de l'Institut Fourier},
title = {Anosov flows and non-Stein symplectic manifolds},
url = {http://eudml.org/doc/75164},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Mitsumatsu, Yoshihiko
TI - Anosov flows and non-Stein symplectic manifolds
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 5
SP - 1407
EP - 1421
AB - 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.
LA - eng
KW - Anosov flows; contact structures; convex symplectic structures
UR - http://eudml.org/doc/75164
ER -

References

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  2. [2] Y. ELIASHBERG, Topological characterization of Stein Manifolds of dimension &gt; 2, International J. Math., 1 (1990), 19-46. Zbl0699.58002
  3. [3] Y. ELIASHBERG, Contact 3-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier, Grenoble, 42 (1-2) (1991), 165-192. Zbl0756.53017MR93k:57029
  4. [4] Y. ELIASHBERG and M. GROMOV, Convex symplectic manifolds, Proc. Symp. Pure Math. A.M.S., 52 (2) (1991), 135-162. Zbl0742.53010MR93f:58073
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  10. [10] M. HANDEL and W.P. THURSTON, Anosov flows on new 3-manifolds, Invent. Math., 59 (1980), 95-103. Zbl0435.58019MR81i:58032
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  12. [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). Zbl0853.57013
  13. [13] D. MCDUFF, Examples of simply-connected non-Kählerian manifolds, J. Diff. Geom., 20 (1984), 267-277. Zbl0567.53031MR86c:57036
  14. [14] D. MCDUFF, Symplectic manifolds with contact type boundaries, Invent. Math., 103 (1991), 651-671. Zbl0719.53015MR92e:53042
  15. [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). Zbl0811.32011
  16. [16] W.P. THURSTON, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc., 55 (1976), 467-468. Zbl0324.53031MR53 #6578
  17. [17] A. WEINSTEIN, On the hypotheses of Rabinowtz's periodic orbit theorems, J. Diff. Eq., 33 (1979), 353-358. Zbl0388.58020MR81a:58030b

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