Journal of the
Korean Mathematical Society JKMS

In this paper, for a $m$-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 $|\textbf{H}_{f}|$ and the weighted real-valued function $f$, we can obtain:
several one-end theorems for $M$;
two Liouville theorems for harmonic maps from $M$ to complete Riemannian manifolds with nonpositive sectional curvature.

Keywords: $L^{p}$ harmonic 1-forms; Submanifolds; Ends; Sectional curvature; $k$-th Ricci curvature.

MSC numbers: 53C20, 53C42.