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Harmonic diffeomorphisms of the hyperbolic plane. (English) Zbl 0796.58014

Starting with an immersion (resp., a diffeomorphism) ϕ:D()D() of the boundary at of the Poincaré model D of the hyperbolic plane, the author finds a harmonic extension (resp., a harmonic diffeomorphism) u:DD. Moreover, u is ± holomorphic (resp., conformal) iff ϕ is conformal. The proof proceeds by construction of a suitable barrier map at D() associated to each ϕ – 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 MH 2 .

Related results for D m D n have been obtained by P. Li and L.-F. Tam [Invent. Math. 105, No. 1, 1-46 (1991; Zbl 0748.58006)].


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