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

DOI:
https://doi.org/10.1090/S0002-9939-1982-0647905-3

MathSciNet review:
647905

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Abstract: Suppose and are complete Riemannian manifolds; with Ricci curvature bounded below by , , with sectional curvature bounded above by a positive constant . Let be a harmonic map such that . If lies inside the cut locus of and , then the energy density of is bounded by a constant depending only on , and . If , then is a constant map.

**[1]**S.-Y. Cheng,*Liouville theorem for harmonic maps*, Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc., Providence, R. I., 1980, pp. 147-151. MR**573431 (81i:58021)****[2]**R. Greene and H. Wu,*Function theory of manifolds which possess a pole*, Lecture Notes in Math., vol. 699, Springer-Verlag, Berlin and New York, 1979. MR**521983 (81a:53002)****[3]**S. Hildebrandt, H. Kaul and K.-O. Widman,*An existence theorem for harmonic mappings of Riemannian manifolds*, Acta Math.**138**(1977), 1-16. MR**0433502 (55:6478)****[4]**S.-T. Yau,*Harmonic functions on complete Riemannian manifolds*, Comm. Pure Appl. Math.**28**(1975), 201-228. MR**0431040 (55:4042)**

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DOI:
https://doi.org/10.1090/S0002-9939-1982-0647905-3

Article copyright:
© Copyright 1982
American Mathematical Society