On complete manifolds supporting a weighted Sobolev type inequality

Levi Adriano and Changyu Xia

Chaos, Solitons & Fractals, 2011, vol. 44, issue 11, 940-946

Abstract: This paper studies the geometric and topological properties of complete open Riemannian manifolds which support a weighted Sobolev or log-Sobolev inequality. We show that the constant in the weighted Sobolev inequality on a complete open Riemannian manifold should be bigger than or equal to the optimal one on the Euclidean space of the same dimension and that a complete open manifold of asymptotically non-negative Ricci curvature supporting a weighted Sobolev inequality must have large volume growth. We also show that a complete manifold of non-negative Ricci curvature on which the log-Sobolev inequality holds is not very far from the Euclidean space.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:44:y:2011:i:11:p:940-946

DOI: 10.1016/j.chaos.2011.07.010

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