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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Hermitian metrics inducing the Poincaré metric, in the leaves of a singular holomorphic foliation by curves

Authors: A. Lins Neto and J. C. Canille Martins
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2963-2988
MSC (2000): Primary 37F75
Published electronically: February 27, 2004
MathSciNet review: 2052604
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Abstract: In this paper we consider the problem of uniformization of the leaves of a holomorphic foliation by curves in a complex manifold $M$. We consider the following problems: 1. When is the uniformization function $\lambda _{g}$, with respect to some metric $g$, continuous? It is known that the metric $\frac{g}{4\lambda _{g}}$ induces the Poincaré metric on the leaves. 2. When is the metric $\frac{g}{4\lambda _{g}}$ complete? We extend the concept of ultra-hyperbolic metric, introduced by Ahlfors in 1938, for singular foliations by curves, and we prove that if there exists a complete ultra-hyperbolic metric $g$, then $\lambda _{g}$ is continuous and $\frac{g}{4\lambda _{g}}$ is complete. In some local cases we construct such metrics, including the saddle-node (Theorem 1) and singularities given by vector fields with the first non-zero jet isolated (Theorem 2). We also give an example where for any metric $g$, $\frac{g}{4\,\lambda _{g}}$ is not complete (§3.2).

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  • [Ah-1] Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0357743
  • [Ah-2] L. V. Ahlfors: ``An Extension of Schwarz' Lemma"; Trans. Am. Math. Soc., 43 (1938), pp. 359-364.
  • [C] Alberto Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 4, 489–516. MR 1235439
  • [C-G] A. Candel and X. Gómez-Mont, Uniformization of the leaves of a rational vector field, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 4, 1123–1133 (English, with English and French summaries). MR 1359843
  • [C-LN] César Camacho and Alcides Lins Neto, Geometric theory of foliations, Birkhäuser Boston, Inc., Boston, MA, 1985. Translated from the Portuguese by Sue E. Goodman. MR 824240
  • [C-S] César Camacho and Paulo Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. (2) 115 (1982), no. 3, 579–595. MR 657239, 10.2307/2007013
  • [El1] E. L. Lima: ``Grupo Fundamental e espaços de recobrimento"; Projeto Euclides, 1993.
  • [F-K] Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. MR 583745
  • [H-K-M] Masuo Hukuhara, Tosihusa Kimura, and Tizuko Matuda, Equations différentielles ordinaires du premier ordre dans le champ complexe, Publications of the Mathematical Society of Japan, 7. The Mathematical Society of Japan, Tokyo, 1961 (French). MR 0124549
  • [K] Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
  • [LN] Alcides Lins Neto, Simultaneous uniformization for the leaves of projective foliations by curves, Bol. Soc. Brasil. Mat. (N.S.) 25 (1994), no. 2, 181–206. MR 1306560, 10.1007/BF01321307
  • [M-P] Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541
  • [M-R] Jean Martinet and Jean-Pierre Ramis, Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Inst. Hautes Études Sci. Publ. Math. 55 (1982), 63–164 (French). MR 672182
  • [V] Alberto Verjovsky, A uniformization theorem for holomorphic foliations, The Lefschetz centennial conference, Part III (Mexico City, 1984) Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1987, pp. 233–253. MR 893869
  • [P] O. Perron: ``Eine neue Behandlung der ersten Randwertaufgabe für $\Delta u=0$"; Math. Z. 18 (1923), 42-54. FM 49, 340.

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Additional Information

A. Lins Neto
Affiliation: Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, Horto, Rio de Janeiro, Brasil

J. C. Canille Martins
Affiliation: LCMAT-UENF, Campos, Rio de Janeiro, Brasil

Received by editor(s): June 19, 2002
Received by editor(s) in revised form: June 2, 2003
Published electronically: February 27, 2004
Additional Notes: This work was supported by FAPESP
Article copyright: © Copyright 2004 American Mathematical Society