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Harmonic Mappings of Negatively Curved Manifolds

Published online by Cambridge University Press:  20 November 2018

Zvi Har'el
Zvi Har'el*
Affiliation:
Technion, Israel Institute of Technology, Haifa 32000, Israel; University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
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Volume decreasing properties of harmonic mappings of space forms were investigated by S. S. Chern and S. I. Goldberg [3] and the author. In a previous paper [6], a step toward generalization of the results was made proving the following theorem:

Theorem. Let ƒ: M —> N be a harmonic mapping of n-dimensional Riemannian manifolds, with C ≦ 0. Suppose the scalar curvature of M is not less than — S, and the Ricci curvature of N is not greater than —S/n, where S ≧ 0 and S > 0 are constants. Then, if u has a maximum on M,

i.e. ƒ is volume decreasing up to a constant.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 3 , 01 June 1978 , pp. 631 - 637
Copyright
Copyright © Canadian Mathematical Society 1978

References

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