The connectivity at infinity of a manifold and Lq,p-Sobolev inequalities

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 ≤ p, q ≤ 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