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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A generalization of the little theorem of Picard

Author: Nicholas C. Petridis
Journal: Proc. Amer. Math. Soc. 61 (1976), 265-271
MSC: Primary 32H25; Secondary 53C20, 30A70
MathSciNet review: 0427691
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Abstract: The method used by the author in cooperation with S. I. Goldberg and T. Ishihara in the study of harmonic mappings of bounded dilatation (J. Differential Geometry 10 (4) (1975)) is employed here to derive a Picard type theorem for quasiconformal mappings into spaces which are not necessarily hyperbolic. More precisely: if $N$ is a nonpositively curved Riemannian manifold of dimension $n$ and scalar curvature bounded away from zero, and if $M$ is a complete locally flat Riemannian manifold of dimension $m \geqslant n$, then every harmonic $K$-quasiconformal mapping $f:M \to N$ is a constant mapping. If, in addition, $M$ and $N$ are Kaehler manifolds then every conformal mapping, $f:M \to N$ is of rank at most $n - 2$. For $m = n = 2$ we obtain the classical “little” theorem of Picard.

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Keywords: Harmonic mappings, quasiconformal mappings, scalar curvature, Picard’s theorem
Article copyright: © Copyright 1976 American Mathematical Society