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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 $u_2 = u_1 \circ \sigma$.

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