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Transactions of the American Mathematical Society

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On the uniform convergence of quasiconformal mappings


Author: Bruce Palka
Journal: Trans. Amer. Math. Soc. 184 (1973), 137-152
MSC: Primary 30A60
DOI: https://doi.org/10.1090/S0002-9947-1973-0340593-4
Erratum: Trans. Amer. Math. Soc. 200 (1974), 445-445.
MathSciNet review: 0340593
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Abstract: Let D be a domain in extended Euclidean n-space with ``smooth'' boundary and let $ \{ {f_j}\} $ be a sequence of K-quasiconformal mappings of D into $ {R^n}$ which converges uniformly on compact sets in D to a quasiconformal mapping. This paper considers the question: When does the sequence $ \{ {f_j}\} $ converge uniformly on all of D? Geometric conditions on the domains $ {f_j}(D)$ are given which are sufficient and, in many cases, necessary for uniform convergence. The particular case where D is the unit ball in $ {R^n}$ is examined to obtain analogues to classical convergence theorems for conformal mappings in the plane.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0340593-4
Keywords: Quasiconformal mapping, uniform convergence, uniform domain, modulus of a path family, Fréchet distance
Article copyright: © Copyright 1973 American Mathematical Society

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