This article concerns the asymptotic Dirichlet problem for harmonic maps into spaces of non-positive sectional curvatures. To be precise, it considers immersions of complete, simply connected Riemannian manifolds \(M\) of non-positive curvature into spaces of non-positive sectional curvatures. The asymptotic Dirichlet problem specifies a continuous boundary value for such a map (on some compactification of \(M\)).
The article first gives a quick new proof of a result of P. Aviles, H. Choi and M. Micallef [J. Funct. Anal. 99, 293-331 (1991; Zbl 0805.53037)], demonstrating the existence and uniqueness of solutions to the asymptotic Dirichlet problem given that the curvature of \(M\) is bounded between two negative constants. It then goes beyond the conditions of this theorem to prove existence and uniqueness when \(M\) is a symmetric space of non-compact type and rank at least two (in which case the sectional curvatures of \(M\) will be at best non-positive).