On complete manifolds supporting a weighted Sobolev type inequality |
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Authors: | Levi Adriano Changyu Xia |
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Institution: | aInstituto de Matemática e Estatística, Universidade Federal de Goiás, 74001-900 Goiânia, GO, Brazil;bDepartamento de Matemática, Universidade de Brasilia, 70910-900 Brasilia, DF, Brazil |
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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. |
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