Gehlawat, S and Verma, K
(2021)
*Two remarks on the PoincarÃ© metric on a singular Riemann surface foliation.*
In: Complex Variables and Elliptic Equations
.

## Abstract

Let (Formula presented.) be a smooth Riemann surface foliation on (Formula presented.), where M is a complex manifold and (Formula presented.) is a closed set. Fix a hermitian metric g on (Formula presented.) and assume that all leaves of (Formula presented.) are hyperbolic. For each leaf (Formula presented.), the ratio of (Formula presented.), the restriction of g to L, and the PoincarÃ© metric (Formula presented.) on L defines a positive function Î· that is known to be continuous on (Formula presented.) under suitable conditions on M, E. For a domain (Formula presented.), we consider (Formula presented.), the restriction of (Formula presented.) to U and the corresponding positive function (Formula presented.) by considering the ratio of g and the PoincarÃ© metric on the leaves of (Formula presented.). First, we study the variation of (Formula presented.) as U varies in the Hausdorff sense motivated by the work of Lins Netoâ��Martins. Secondly, Minda had shown the existence of a domain Bloch constant for a hyperbolic Riemann surface S, which in other words shows that every holomorphic map from the unit disc into S, whose distortion at the origin is bounded below, must be locally injective in some hyperbolic ball of uniform radius. We show how to deduce a version of this Bloch constant for (Formula presented.). Â© 2021 Informa UK Limited, trading as Taylor & Francis Group.

Item Type: | Journal Article |
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Publication: | Complex Variables and Elliptic Equations |

Publisher: | Taylor and Francis Ltd. |

Additional Information: | The copyright for this article belongs to Taylor and Francis Ltd. |

Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |

Date Deposited: | 27 Aug 2021 11:22 |

Last Modified: | 27 Aug 2021 11:22 |

URI: | http://eprints.iisc.ac.in/id/eprint/69568 |

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