Asymptotic behavior of harmonic maps from complete manifolds (Q1302285)

Let \(M\) be an \(m\)-dimensional noncompact Riemannian manifold that possesses finite ends which are all large, i.e., each unbounded component of \(M\setminus D\) satisfies a certain volume increasing condition, where \(D\) is a compact subset of \(M\). Suppose that \(M\) satisfies certain volume comparison condition and its Ricci curvature \(\text{Ric}_M(x)\geq -(m+ 1)K/(1+ r(x))^2\), where \(r(x)\) is the distance from one point \(p\in M\) to \(x\in M\). Furthermore, let \(N\) be a compact manifold with sectional curvature bounded above by a positive constant. The author establishes a finite energy harmonic mapping \(u: M\to N\) with \(u(M)\) contained in a normal geodesic ball in \(N\) such that it must be asymptotically constant at the infinity of each large end of \(M\).