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Proceedings of the American Mathematical Society

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

 

 

Convex functions and harmonic maps


Author: William B. Gordon
Journal: Proc. Amer. Math. Soc. 33 (1972), 433-437
MSC: Primary 53C20
MathSciNet review: 0291987
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Abstract: A subset D of a riemannian manifold Y is said to be convex supporting if every compact subset of D has a Y-open neighborhood which supports a strictly convex function. The image of a harmonic map f from a compact manifold X to Y cannot be contained in any convex supporting subset of Y unless f is constant. Also, if Y has a convex supporting covering space and $ {\pi _1}(X)$ is finite then every harmonic map from X to Y is necessarily constant. Examples of convex supporting domains and manifolds are given.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0291987-1
Keywords: Harmonic map, convex function
Article copyright: © Copyright 1972 American Mathematical Society