The connectivity at infinity of a manifold and $L^{q,p}$-Sobolev inequalities

Abstract

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 ($2\le p$, $q\le p^{\ast }$), 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.

fixed error in previous update (forgot to delete old version that was appended to the new one)

References in corpus (1)

• Sobolev Inequalities for Differential Forms and $L_{q,p}$-cohomology

Cited by in corpus (4)

• Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem
• Stokes' theorem, volume growth and parabolicity
• The Feller property on Riemannian manifolds
• A note on a Sung-Wang's paper