  In this paper we present some extensions of the celebrated finite point
conformal compactification theorem of Huber \cite{Hu57} for complete open
surfaces to general dimensions based on the n-Laplace equations in conformal
geometry. We are able to conclude a domain in the round sphere has to be the
sphere deleted finitely many points if it can be endowed with a complete
conformal metric with the negative part of the smallest Ricci curvature
satisfying some integrable conditions. Our proof is based on the strengthened
version of the Arsove-Huber's type theorem on n-superharmonic functions in our
earlier work \cite{MQ18}. Moreover, using p-parabolicity, we push the
injectivity theorem of Schoen-Yau to allow some negative curvature and
therefore establish the finite point conformal compactification theorem for
manifolds that have a conformal immersion into the round sphere. As a side
product we establish the injectivity of conformal immersions from
n-parabolicity alone, which is interesting by itself in conformal geometry.
