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Transactions of the American Mathematical Society

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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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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

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