HARMONIC TRANSFORMATIONS OF THE HYPERBOLIC PLANE

Abstract

Let (H, g) denote the upper half plane in R2 with the Riemannian metric g := ((dx)2 + (dy)2)/y2. First of all we get a necessary and sufficient condition for a diffeomorphism φ of (H, g) to be a harmonic map. And, we obtain the fact that if a diffeomorphism φ of (H, g) is a harmonic function, then the following facts are equivalent:

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