  The purpose of this note is to study the connectedness at infinity of
manifold by using the theory of $p$-harmonic functions. We show that if the
first eigenvalue $\lambda_{1,p}$ for the $p$-Laplacian achievies its maximal
value on a K\"{a}hler manifold or a quaternionic K\"{a}hler manifold then such
a manifold must be connected at infinity unless it is a topological cylinder
with an explicit warped product metric.
