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Constant mean curvature surface, harmonic maps, and universal Teichmüller space. (English) Zbl 0808.53056
T. K. Milnor [Trans. Am. Math. Soc. 280, 161-185 (1983; Zbl 0532.53047)]has proven that the Gauss map of a space-like surface in Minkowski three-space ${M}^{2,1}$ is harmonic if and only if the mean curvature of the surface is constant. Using this fact, the author constructs, for each holomorphic quadratic differential on a simply- connected domain ${\Omega }$ in $ℂ$, an injective harmonic map from ${\Omega }$ into the Poincaré disc $D$. This harmonic map is unique up to equivalent classes, provided it satisfies a completeness condition. Also, a classification of all complete hyperbolic space-like surfaces of constant mean curvature in ${M}^{2,1}$ has been found. M. Wolf [J. Differ. Geom. 29, No. 2, 449-479 (1989; Zbl 0655.58009)]shows that harmonic maps between surfaces are deeply related to the Teichmüller theory of compact Riemann surfaces. The author studies the analogy for the universal Teichmüller space and proves that a harmonic diffeomorphism from $D$ onto itself is quasi-conformal if and only if the associated Hopf differential is bounded with respect to the Poincaré metric. As a consequence, a continuous map from the space of equivalence classes of holomorphic quadratic differentials under the action of the Möbius group, which is bounded with respect to the Poincaré metric, into the universal Teichmüller space has been obtained.

##### MSC:
 53C42 Immersions (differential geometry) 58E20 Harmonic maps between infinite-dimensional spaces