Fatou property on harmonic maps from complete manifolds with nonnegative curvature at infinity into convex balls (Q1899640)

Consider a Riemannian manifold \(N\) and let \(B_\tau (p)\) denote a geodesic ball in \(N\). The author then discusses harmonic maps from \(M\) into \(B_\tau (p)\), where \(M\) is a complete noncompact Riemannian manifold with nonnegative sectional curvature at infinity which means that there exists a compact subset \(D\) of \(M\) such that the sectional curvature on \(M - D\) is nonnegative. The following theorem is demonstrated: Let \(E_1, \dots, E_l\), \(l \geq 2\), denote the ends of \(M\) and fix points \(p_1, \dots, p_l\) in \(B_\tau (p)\). Then there exists a unique harmonic map \(u : M \to B_\tau (p)\) with \(u(x) \to p_k\) as \(x \in E_k\), \(x \to \infty\), \(k = 1, \dots, l\), the energy of which is finite. Further sections study Fatou's property in detail, for example, it is shown that for any harmonic map \(M \to B_\tau(p)\) with finite energy there exist points \(p_1, \dots, p_l \in B_\tau (p)\) such that the asymptotic behaviour described above is true.