Harmonic maps between unbounded convex polyhedra in hyperbolic spaces (Q1320490)

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 \textit{P. Li} and \textit{L.-F. Tam} [Ann. Math., II. Ser. 137, No. 1, 167-201 (1993; Zbl 0776.58010)].