The connectivity at infinity of a manifold and 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,L-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,L-p-Sobolev inequality. (C) 2013 Elsevier GmbH. All rights reserved.