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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 \(\Phi\) can be realized as the Hopf-differential of a harmonic diffeomorphism. They get the following existence theorem and uniqueness theorem:
Existence-theorem: Let \(N = (D, e^{2\psi} ds^2_p)\) be a hyperbolic CH-surface with Gaussian curvature \(K_N\) satisfying \(-b^2 \leq K_N \leq 0\) for some constant \(b > 0\) and \(\lambda_1(N) > 0\). Then given any holomorphic quadratic differential \(\Phi = \phi dz^2\) on \(D(R_0)\), \(R_0 = 1\) or \(\infty\), there is a harmonic map \(u\) from \(D(R_0)\) to \(N\) with Hopf differential given by \(\Phi\). Moreover, if \(R_0 = 1\) or \(\phi\) is not a constant, then \(u\) can be chosen to be a harmonic diffeomorphism into \(N\). Furthermore, if \(R_0 = 1\) and \(\Phi \in BDQ(D)\), then \(u\) can be chosen to be a quasi-conformal harmonic diffeomorphism onto \(N\).
Uniqueness-theorem: Let \(H = (D,ds^2_p)\) be the Poincaré disk and let \(N\) be a hyperbolic CH-surface with Gaussian curvature \(K_N\). Let \(\phi dz^2\) 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 \(\phi dz^2\). Suppose that \(\text{exp}(2\omega_i)ds^2_p\) is complete on \(D\) for \(i = 1,2\), where \(\omega_i =\log |\partial u_i|\), and suppose \(K_N(u_1(z))= K_N(u_2(z))\) for all \(z \in D\). Then there is an isometry \(\sigma : N \to N\) such that \(u_2 = u_1 \circ \sigma\).

MSC:

58E20 Harmonic maps, etc.
57R50 Differential topological aspects of diffeomorphisms
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