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]**Shiu Yuen Cheng,*Liouville theorem for harmonic maps*, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 147–151. MR**573431****[2]**R. E. Greene and H. Wu,*Function theory on manifolds which possess a pole*, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. MR**521983****[3]**Stéfan Hildebrandt, Helmut Kaul, and Kjell-Ove Widman,*An existence theorem for harmonic mappings of Riemannian manifolds*, Acta Math.**138**(1977), no. 1-2, 1–16. MR**0433502**, https://doi.org/10.1007/BF02392311**[4]**Shing Tung Yau,*Harmonic functions on complete Riemannian manifolds*, Comm. Pure Appl. Math.**28**(1975), 201–228. MR**0431040**, https://doi.org/10.1002/cpa.3160280203

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

Article copyright:
© Copyright 1982
American Mathematical Society