We consider complete manifolds with asymptotically non-negative curvature which enjoy a Euclidean-type Sobolev inequality and we get an explicit lower control on the volume of geodesic balls. In case the amount of negative curvature is small and the Sobolev constant is almost optimal, we deduce that the manifold is diffeomorphic to Euclidean space. This extends previous results by M. Ledoux and C. Xia. © 2010 American Mathematical Society
Pigola, S., Veronelli, G. (2010). Lower volume estimates and sobolev inequalities. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 138(12), 4479-4486 [10.1090/S0002-9939-2010-10514-2].
Lower volume estimates and sobolev inequalities
Pigola, S;Veronelli, G
2010
Abstract
We consider complete manifolds with asymptotically non-negative curvature which enjoy a Euclidean-type Sobolev inequality and we get an explicit lower control on the volume of geodesic balls. In case the amount of negative curvature is small and the Sobolev constant is almost optimal, we deduce that the manifold is diffeomorphic to Euclidean space. This extends previous results by M. Ledoux and C. Xia. © 2010 American Mathematical Society
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