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Proceedings of the American Mathematical Society

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

 

 

On the harmonic maps from $ {\bf R}\sp 2$ into $ H\sp 2$


Author: Jun Min Lin
Journal: Proc. Amer. Math. Soc. 108 (1990), 521-527
MSC: Primary 58E20; Secondary 30C60, 35Q99
DOI: https://doi.org/10.1090/S0002-9939-1990-0975649-1
MathSciNet review: 975649
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Abstract: In this paper, we prove that normalized harmonic maps from $ {{\mathbf{R}}^2}$ or $ {{\mathbf{R}}^2}\backslash \{ 0\} $ into $ {H^2}$ are just geodesies on $ {H^2}$ and that the quasiconformal harmonic maps from $ {{\mathbf{R}}^2}$ into $ {H^2}$ are constant maps. We prove also that the only solution to $ \Delta \alpha = \sinh \alpha \,$ on $ {{\mathbf{R}}^2}\backslash \{ 0\} $ is the zero solution.


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DOI: https://doi.org/10.1090/S0002-9939-1990-0975649-1
Article copyright: © Copyright 1990 American Mathematical Society