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Mappings of nonpositively curved manifolds

Published online by Cambridge University Press:  22 January 2016

Samuel I. Goldberg
Samuel I. Goldberg*
Affiliation:
University of Illinois
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In recent papers with S. S. Chern [3] and T.Ishihara [4], the author studied both the volume—and distance—decreasing properties of harmonic mappings thereby obtaining real analogues and generalizations of the classical Schwarz-Ahlfors lemma, as well as Liouville’s theorem and the little Picard theorem. The domain M in the first case was the open ball with the hyperbolic metric of constant negative curvature, and the target was a negatively curved Riemannian manifold with sectional curvature bounded away from zero. In this paper, it is shown that M may be taken to be any complete Riemannian manifold of non-positive curvature.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 61 , July 1976 , pp. 73 - 84
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1976

References

[1] Bishop, R. L. and Crittenden, R. J., Geometry of Manifolds, Academic Press Inc., New York, 1964.Google Scholar
[2] Bishop, R. L. and O’Neil, B., Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49.CrossRefGoogle Scholar
[3] Chern, S. S. and Goldberg, S. I., On the volume-decreasing property of a class of real harmonic mappings, Amer. J. Math. 97 (1975), 133–147.Google Scholar
[4] Goldberg, S. I. and Ishihara, T., Harmonic quasiconformal mappings of Riemannian manifolds, Bull. Amer. Math. Soc. 80 (1974), 562–566.Google Scholar
[5] Goldberg, S. I. and Ishihara, T., Harmonic quasiconformal mappings of Riemannian manifolds, Amer. J. Math., to appear.Google Scholar
[6] Goldberg, S. I., Ishihara, T. and Petridis, N. C., Mappings of bounded dilatation, Journ. of Math., Tokushima Univ. 8 (1974), 1–7.Google Scholar
[7] Har’El, Z., Harmonic mappings and distortion theorems, thesis, Technion, Israel Inst. of Tech., Haifa, 1975.Google Scholar
[8] Kiernan, P. J., Quasiconformal mappings and Schwarz’s lemma, Trans. Amer. Math. Soc. 148 (1970), 185–197.Google Scholar