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In this paper we present some extensions of the celebrated finite pointconformal compactification theorem of Huber \cite{Hu57} for complete opensurfaces to general dimensions based on the n-Laplace equations in conformalgeometry. We are able to conclude a domain in the round sphere has to be thesphere deleted finitely many points if it can be endowed with a completeconformal metric with the negative part of the smallest Ricci curvaturesatisfying some integrable conditions. Our proof is based on the strengthenedversion of the Arsove-Huber's type theorem on n-superharmonic functions in ourearlier work \cite{MQ18}. Moreover, using p-parabolicity, we push theinjectivity theorem of Schoen-Yau to allow some negative curvature andtherefore establish the finite point conformal compactification theorem formanifolds that have a conformal immersion into the round sphere. As a sideproduct we establish the injectivity of conformal immersions fromn-parabolicity alone, which is interesting by itself in conformal geometry.

On Huber's type theorems in general dimensions