# Anosov flows and non-Stein symplectic manifolds

Annales de l'institut Fourier (1995)

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

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topMitsumatsu, 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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- [9] R.C. GUNNING and H. ROSSI, Analytic functions of several complex variables, Prentice-Hall Inc., Englewood Cliffs N.J., 1965. Zbl0141.08601MR31 #4927
- [10] M. HANDEL and W.P. THURSTON, Anosov flows on new 3-manifolds, Invent. Math., 59 (1980), 95-103. Zbl0435.58019MR81i:58032
- [11] M. HIRSCH, C. Pugh and M. SHUB, Invariant manifolds, Springer Lecture Notes in Mathematics, 583 (1977). Zbl0355.58009MR58 #18595
- [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] D. MCDUFF, Examples of simply-connected non-Kählerian manifolds, J. Diff. Geom., 20 (1984), 267-277. Zbl0567.53031MR86c:57036
- [14] D. MCDUFF, Symplectic manifolds with contact type boundaries, Invent. Math., 103 (1991), 651-671. Zbl0719.53015MR92e:53042
- [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] W.P. THURSTON, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc., 55 (1976), 467-468. Zbl0324.53031MR53 #6578
- [17] A. WEINSTEIN, On the hypotheses of Rabinowtz's periodic orbit theorems, J. Diff. Eq., 33 (1979), 353-358. Zbl0388.58020MR81a:58030b

## Citations in EuDML Documents

top- David E. Blair, Special directions on contact metric manifolds of negative $\xi $-sectional curvature
- Takeo Noda, Projectively Anosov flows with differentiable (un)stable foliations
- Takeo Noda, Regular projectively Anosov flows with compact leaves
- Masayuki Asaoka, Regular projectively Anosov flows on three-dimensional manifolds

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