*(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 $N=(D,{e}^{2\psi}d{s}_{p}^{2})$ be a hyperbolic CH-surface with Gaussian curvature ${K}_{N}$ satisfying $-{b}^{2}\le {K}_{N}\le 0$ for some constant $b>0$ and ${\lambda}_{1}\left(N\right)>0$. Then given any holomorphic quadratic differential ${\Phi}=\phi d{z}^{2}$ on $D\left({R}_{0}\right)$, ${R}_{0}=1$ or $\infty $, there is a harmonic map $u$ from $D\left({R}_{0}\right)$ 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 ${\Phi}\in BDQ\left(D\right)$, then $u$ can be chosen to be a quasi-conformal harmonic diffeomorphism onto $N$.

Uniqueness-theorem: Let $H=(D,d{s}_{p}^{2})$ be the PoincarĂ© disk and let $N$ be a hyperbolic CH-surface with Gaussian curvature ${K}_{N}$. Let $\phi d{z}^{2}$ 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 $\phi d{z}^{2}$. Suppose that $\text{exp}\left(2{\omega}_{i}\right)d{s}_{p}^{2}$ is complete on $D$ for $i=1,2$, where ${\omega}_{i}=log\left|\partial {u}_{i}\right|$, and suppose ${K}_{N}\left({u}_{1}\left(z\right)\right)={K}_{N}\left({u}_{2}\left(z\right)\right)$ for all $z\in D$. Then there is an isometry $\sigma :N\to N$ such that ${u}_{2}={u}_{1}\circ \sigma $.