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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the Liouville theorem for harmonic maps


Author: Hyeong In Choi
Journal: Proc. Amer. Math. Soc. 85 (1982), 91-94
MSC: Primary 53C99; Secondary 58E20
MathSciNet review: 647905
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Abstract: Suppose $ M$ and $ N$ are complete Riemannian manifolds; $ M$ with Ricci curvature bounded below by $ - A$, $ A \geqslant 0$, $ N$ with sectional curvature bounded above by a positive constant $ K$. Let $ u:M \to N$ be a harmonic map such that $ u(M) \subset {B_R}({y_0})$. If $ {B_R}({y_0})$ lies inside the cut locus of $ {y_0}$ and $ R < \pi /2\sqrt K $, then the energy density $ e(u)$ of $ u$ is bounded by a constant depending only on $ A$, $ K$ and $ R$. If $ A = 0$, then $ u$ is a constant map.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1982-0647905-3
PII: S 0002-9939(1982)0647905-3
Article copyright: © Copyright 1982 American Mathematical Society