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Harmonic maps from the complex plane into surfaces with nonpositive curvature. (English) Zbl 0843.58028
Summary: We give a characterization for an orientation preserving harmonic diffeomorphism from $\bbfC$ into a complete, simply connected, negatively pinched surface to have a polynomial growth Hopf differential. In particular, we prove that an orientation preserving harmonic diffeomorphism from $\bbfC$ into the Poincaré disk $\bbfH$ has a polynomial growth Hopf differential of degree $m$ if and only if its image is an ideal polygon with $m+2$ vertices on $\partial \bbfH$, with the assumption that the conformal metric on $\bbfC$ with the $\partial$-energy density as the conformal factor is complete. We will describe the geometric behavior of this harmonic diffeomorphisms in terms of the trajectories of their Hopf differentials. We will also construct all harmonic diffeomorphisms in this class, and prove that there is an $m-1$ parameter family of nontrivially distinct harmonic diffeomorphisms from the complex plane to a fixed ideal polygon with $m + 2$ vertices in the hyperbolic plane. In particular, such harmonic maps are not unique, answering a question of Schoen.

58E20Harmonic maps between infinite-dimensional spaces
53C42Immersions (differential geometry)
53C50Lorentz manifolds, manifolds with indefinite metrics