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 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