  The purpose of this paper is to give a self-contained proof that a complete
manifold with more than one end never supports an $L^{q,p}$-Sobolev inequality
($2 \leq p$, $q\leq p^{*}$), provided the negative part of its Ricci tensor is
small (in a suitable spectral sense). In the route, we discuss potential
theoretic properties of the ends of a manifold enjoying an $L^{q,p}$-Sobolev
inequality.
