Boundary behavior of functions on complete manifolds of negative curvature (Q1262409)

We study harmonic and h-harmonic functions on complete negatively curved manifolds, and their behavior near its sphere at infinity. Let M be a complete, simply connected Riemannian manifolds, which sectional curvatures are bounded by two negative constants. For such manifold, \textit{M. T. Anderson} and \textit{R. Schoen} proved that the Martin compactification of M is homeomorphic to one of Eberlein and O'Neill, and that positive harmonic functions on M satisfy the so-called Fatou phenomena [Ann. Math., II. Ser. 121, 429-461 (1985; Zbl 0587.53045)]. In our paper we introduce a class of approach regions and prove local Fatou phenomena of harmonic and h-harmonic functions on M. Furthermore, we investigate their fine, semifine, nontangential and admissible limits at the sphere at infinity. Our results extend also Theorem 4 in \textit{A. Ancona} [Ann. Math., II. Ser. 125, 495-536 (1987; Zbl 0652.31008)].