The connectivity at infinity of a manifold and Lq,pLq,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 Lq,pLq,p-Sobolev inequality (2≤p2≤p, q≤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 Lq,pLq,p-Sobolev inequality.