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Harmonic diffeomorphisms into Cartan-Hadamard surfaces with prescribed Hopf differentials. (English) Zbl 0842.58014

A complete simply connected surface with Gaussian curvature bounded between a negative constant and 0 is called hyperbolic Cartan-Hadamard surface (or simply hyperbolic CH-surface). In this paper the authors study the problem when a given holomorphic quadratic differential form Φ can be realized as the Hopf-differential of a harmonic diffeomorphism. They get the following existence theorem and uniqueness theorem:

Existence-theorem: Let N=(D,e 2ψ ds p 2 ) be a hyperbolic CH-surface with Gaussian curvature K N satisfying -b 2 K N 0 for some constant b>0 and λ 1 (N)>0. Then given any holomorphic quadratic differential Φ=φdz 2 on D(R 0 ), R 0 =1 or , there is a harmonic map u from D(R 0 ) to N with Hopf differential given by Φ. Moreover, if R 0 =1 or φ is not a constant, then u can be chosen to be a harmonic diffeomorphism into N. Furthermore, if R 0 =1 and ΦBDQ(D), then u can be chosen to be a quasi-conformal harmonic diffeomorphism onto N.

Uniqueness-theorem: Let H=(D,ds p 2 ) be the PoincarĂ© disk and let N be a hyperbolic CH-surface with Gaussian curvature K N . Let φdz 2 be a holomorphic quadratic differential in BDQ(D). Let u 1 and u 2 be two orientation preserving harmonic diffeomorphisms from H into N with the same Hopf differential φdz 2 . Suppose that exp(2ω i )ds p 2 is complete on D for i=1,2, where ω i =log|u i |, and suppose K N (u 1 (z))=K N (u 2 (z)) for all zD. Then there is an isometry σ:NN such that u 2 =u 1 σ.

MSC:
58E20Harmonic maps between infinite-dimensional spaces
57R50Diffeomorphisms (differential topology)