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Generalizations of Picard's theorem for Riemann surfaces


Author: Pentti Järvi
Journal: Trans. Amer. Math. Soc. 323 (1991), 749-763
MSC: Primary 30F25; Secondary 30D40
DOI: https://doi.org/10.1090/S0002-9947-1991-1030508-9
MathSciNet review: 1030508
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Abstract: Let $ D$ be a plane domain, $ E \subset D$ a compact set of capacity zero, and $ f$ a holomorphic mapping of $ D\backslash E$ into a hyperbolic Riemann surface $ W$. Then there is a Riemann surface $ W'$ containing $ W$ such that $ f$ extends to a holomorphic mapping of $ D$ into $ W'$. The same conclusion holds if hyperbolicity is replaced by the assumption that the genus of $ W$ be at least two. Furthermore, there is quite a general class of sets of positive capacity which are removable in the above sense for holomorphic mappings into Riemann surfaces of positive genus, except for tori.


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DOI: https://doi.org/10.1090/S0002-9947-1991-1030508-9
Article copyright: © Copyright 1991 American Mathematical Society

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