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Harmonic maps between unbounded convex polyhedra in hyperbolic spaces. (English) Zbl 0812.58017

The authors solve the Dirichlet problem for harmonic maps between two convex and unbounded polyhedra \(\Omega_ 1\) and \(\Omega_ 2\) with finitely many sides in hyperbolic spaces \(H^ m\) and \(H^ n\) respectively. The boundary maps are supposed to satisfy some natural conditions like respecting divisions on faces belonging or not to a sphere at infinity.
For \(n = m = 2\) with some additional conditions the existence of harmonic diffeomorphisms is proved.
To get the solution the authors construct a nice barrier map for a given boundary map following P. Li and L.-F. Tam [Ann. Math., II. Ser. 137, No. 1, 167-201 (1993; Zbl 0776.58010)].

MSC:

58E20 Harmonic maps, etc.
58J32 Boundary value problems on manifolds

Citations:

Zbl 0776.58010
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References:

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