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Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space. (English) Zbl 0717.53038
Authors’ abstract: “The Gauss map of a spacelike constant mean curvature hypersurface of Minkowski space is a harmonic map to hyperbolic space. The properties of such hypersurfaces are interpreted in terms of the harmonic mapping. Given an arbitrary closed set in the ideal boundary at infinity of hyperbolic space, there exists a complete entire spacelike constant mean curvature hypersurface whose Gauss map is a diffeomorphism onto the interior of the hyperbolic space convex hull of the set. Identifying ideal infinity with the light cone, this set corresponds to the lightlike directions of the hypersurface. In terms of this intrinsic data we give conditions for the hyperbolicity or parabolicity of this hypersurface. For example, if the set of lightlike directions has nonempty interior in the unit sphere, then this hypersurface can be constructed so as to admit nontrivial bounded harmonic functions. This gives many new examples of harmonic maps of the disk and the complex plane to the hyperbolic plane, which are of full rank.”
Reviewer: B.Palmer

MSC:
53C42Immersions (differential geometry)
58E20Harmonic maps between infinite-dimensional spaces