An obstruction to the conformal compactification of Riemannian manifolds

Abstract

In this paper, we study noncompact complete Riemannian $n$-manifolds with $n\ge 3$ which are not pointwise conformal to subdomains of any compact Riemannian $n$-manifold. For this, we compare the Sobolev Quotient at infinity of a noncompact complete Riemannian manifold with that of the singular set in a compact Riemannian manifold using the method for the Yamabe problem.