Uniformization of the leaves of a rational vector field
Candel, Alberto ; Gómez-Mont, X.
Annales de l'Institut Fourier, Tome 45 (1995), p. 1123-1133 / Harvested from Numdam
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Nous étudions la structure analytique des feuilles d’un feuilletage holomorphe par des courbes dans une variété complexe. Nous montrons que si chaque feuille est une surface hyperbolique, alors l’application d’uniformisation est continue. Dans le cas de l’espace projectif complexe il suffit qu’il n’y ait pas de feuille algébrique.

We study the analytic structure of the leaves of a holomorphic foliation by curves on a compact complex manifold. We show that if every leaf is a hyperbolic surface then they can be simultaneously uniformized in a continuous manner. In case the manifold is complex projective space a sufficient condition is that there are no algebraic leaf.

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     author = {Candel, Alberto and G\'omez-Mont, X.},
     title = {Uniformization of the leaves of a rational vector field},
     journal = {Annales de l'Institut Fourier},
     volume = {45},
     year = {1995},
     pages = {1123-1133},
     doi = {10.5802/aif.1488},
     mrnumber = {96k:32068},
     zbl = {0832.32017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1995__45_4_1123_0}
}

Candel, Alberto; Gómez-Mont, X. Uniformization of the leaves of a rational vector field. Annales de l'Institut Fourier, Tome 45 (1995) pp. 1123-1133. doi : 10.5802/aif.1488. http://gdmltest.u-ga.fr/item/AIF_1995__45_4_1123_0/

[1]L. Ahlfors, Complex Analysis, MacGraw Hill, New York, 1966. | Zbl 0154.31904

[2]L. Ahlfors, Conformal Invariants, MacGraw Hill, New York, 1973.

[3]R. Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc., 235 (1978), 213-219. | MR 57 #10010 | Zbl 0416.32013

[4]C. Camacho, A. Lins, P. Sad, Minimal sets of foliations on complex projective spaces, Publ. Math. IHES, 68 (1989), 187-203. | Numdam | Zbl 0682.57012

[5]A. Candel, Uniformization of surface laminations, Ann. Scient. Ec. Norm. Sup., 26 (1993), 489-516. | Numdam | MR 94f:57025 | Zbl 0785.57009

[6]D. Cerveau, Equations différentielles algébriques : Remarques et problèmes, J. Fac. Sci. Univ. Tokyo, 36 (1989), 665-680. | MR 91b:32035 | Zbl 0698.58047

[7]E. Ghys, Gauss-Bonnet Theorem for 2-dimensional foliations, J. of Funct. Anal., 77 (1988), 51-59. | MR 89d:57040 | Zbl 0656.57017

[8]A.A. Glutsuk, The hyperbolicity of phase curves of a generic polynomial vector field in Cn, Functional Analysis and its applications, 28 (1994), 77-84. | Zbl 0848.32032

[9]A. Lins, Simultaneous uniformization for the leaves of projective foliations by curves, to appear in Bol. Soc. Brasileira. | Zbl 0821.32027

[10]C. Moore & C. Schochet, Global analysis of foliated spaces, Springer-Verlag, New York, 1988. | Zbl 0648.58034

[11]A. Verjovsky, A uniformization theorem for holomorphic foliations., Contemp. Math., 58(III) (1987), 233-253. | MR 88h:57027 | Zbl 0619.32017