arXiv 1007.17612010-07math.DG

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

Stefano Pigola, Alberto G. Setti, Marc Troyanov

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英文摘要：

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 ($2 \leq p$, $q\leq p^{*}$), 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.

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