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Harmonic diffeomorphisms of the hyperbolic plane. (English) Zbl 0796.58014
Starting with an immersion (resp., a diffeomorphism) $\varphi:D(\infty) \to D (\infty)$ of the boundary at $\infty$ of the Poincaré model $D$ of the hyperbolic plane, the author finds a harmonic extension (resp., a harmonic diffeomorphism) $u:D \to D$. Moreover, $u$ is $\pm$ holomorphic (resp., conformal) iff $\varphi$ is conformal. The proof proceeds by construction of a suitable barrier map at $D(\infty)$ associated to each $\varphi$ -- and to obtain from it an a priori growth estimate. He also constructs entire spacelike constant mean curvature surfaces $M$ in Minkowski 3-space whose Gauss maps are harmonic diffeomorphisms $M \to H\sp 2$. Related results for $D\sp m \to D\sp n$ have been obtained by {\it P. Li} and {\it L.-F. Tam} [Invent. Math. 105, No. 1, 1-46 (1991; Zbl 0748.58006)].

58E20Harmonic maps between infinite-dimensional spaces
30C20Conformal mappings of special domains
53C42Immersions (differential geometry)
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