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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A nullset for normal functions in several variables


Author: Juhani Riihentaus
Journal: Proc. Amer. Math. Soc. 110 (1990), 923-933
MSC: Primary 32A17; Secondary 32D20
MathSciNet review: 1028048
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Abstract: Suppose that $ \Omega $ is a domain in $ {C^n},E \subset \Omega $ is closed in $ \Omega $, and $ f:\Omega \backslash E \to {C^ * }$ is a meromorphic function. We show that if $ f$ is normal and $ E$ is an analytic subvariety or, more generally, of locally finite $ (2n - 2)$-dimensional Hausdorff measure in $ \Omega $ satisfying a certain geometric condition, then $ f$ can be extended to a meromorphic function (= holomorphic mapping) $ {f^ * }:\Omega \to {C^ * }$. In the case of a subvariety sufficient, but not necessary, for the geometric condition is that the singularities of $ E$ are normal crossings. As a digression, we give a new proof for the following result, due to Parreau in the case $ n = 1$: if $ f$ is in the Nevanlinna class and $ E$ is polar (in $ {R^{2n}}$), then $ f$ has a meromorphic extension $ {f^ * }$ to $ \Omega $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1028048-0
PII: S 0002-9939(1990)1028048-0
Keywords: Normal function, Nevanlinna class
Article copyright: © Copyright 1990 American Mathematical Society