On the Harmonic Maps from R 2 into H 2

Abstract

In this paper, we prove that normal.zed harmonic maps from R2 or R2\{O} into H2 are just geodesics on H2 and that the quasiconformal harmonic maps from R2 into H2 are constant maps. We prove also that the only solution to Aa = sinh a on R2\{O} is the zero solution. INTRODUCTION Harmonic map is a common generalization of minimum submanifolds, harmonic functions and nonlinear a-models ([3], [4]). We have a better understanding of harmonic maps from compact manifolds ([3], [5]), whereas about harmonic maps from noncompact manifolds we know rather little. Hence, it is very interesting to know harmonic from R2 into H2 ([4]). Here, we give a description of harmonic maps from R2 into H2 under some additional assumptions. Theorem 1. The only normalized harmonic map (o from R2 or R 2\{} into H is geodesic, i.e., under suitable coordinate system, o(x,y) = y(x), where y:R H2 is a geodesic. The concept of normalized harmonic map is introduced in [7] and [8]. It is known that normalized harmonic maps from Q c R2 into H2 exist locally ([7]). Our theorem shows however, that the global problem is quite different from the local one. (The definition of normalized harmonic map will be given in ?2.) The proof of the theorem is reduced to the discussion of the solution to the Sinh-Laplace equation: Aa = sinh a on R2 or R2\{} . The local version of the reduction is given by [7], and a global version will be given in ?2. Received by the editors October 4, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 58E20. (3 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page