Hermitian metrics inducing the Poincaré metric, in the leaves of a singular holomorphic foliation by curves
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- by A. Lins Neto and J. C. Canille Martins
- Trans. Amer. Math. Soc. 356 (2004), 2963-2988
- DOI: https://doi.org/10.1090/S0002-9947-04-03434-8
- Published electronically: February 27, 2004
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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).References
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Bibliographic Information
- A. Lins Neto
- Affiliation: Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, Horto, Rio de Janeiro, Brasil
- Email: alcides@impa.br
- J. C. Canille Martins
- Affiliation: LCMAT-UENF, Campos, Rio de Janeiro, Brasil
- Email: canille@uenf.br
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 2963-2988
- MSC (2000): Primary 37F75
- DOI: https://doi.org/10.1090/S0002-9947-04-03434-8
- MathSciNet review: 2052604