The topology at infinity of a manifold supporting an $L^{q,p}$-Sobolev inequality

Details The topology at infinity of a manifold supporting an $L^{q,p}$-Sobolev inequality The topology at infinity of a manifold supporting an $L^{q,p}$-Sobolev inequality

2010-07-11

arxiv.org/abs/1007.1761v1

Mathematics

Differential Geometry

Scientific paper

This note aims to prove that a complete manifold supporting a general $L^{q,p}$-Sobolev inequality is connected at infinity provided the negative part of its Ricci tensor is small (in a suitable spectral sense). In the route, we deduce potential theoretic and volume properties of the ends of a manifold enjoying the $L^{q,p}$-Sobolev inequality. Our results are related to previous work by I. Holopainen and S.W. Kim and Y.H. Lee.

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