Journal of the
Korean Mathematical Society JKMS

In this paper, for an $m$-dimensional ($m\geq5$) complete noncompact submanifold $M$ immersed in an $n$-dimensional ($n\geq6$) simply connected Riemannian manifold $N$ with negative sectional curvature, under suitable constraints on the squared norm of the second fundamental form of $M$, the norm of its weighted mean curvature vector $|\textbf{\emph{H}}_{f}|$ and the weighted real-valued function $f$, we can obtain:

$\bullet$ several one-end theorems for $M$; 

$\bullet$ two Liouville theorems for harmonic maps from $M$ to complete Riemannian manifolds with nonpositive sectional curvature.