Lower volume estimates and Sobolev inequalities

Pigola, Stefano; Veronelli, Giona

doi:10.1090/S0002-9939-2010-10514-2

Lower volume estimates and Sobolev inequalities
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by Stefano Pigola and Giona Veronelli PDF

Proc. Amer. Math. Soc. 138 (2010), 4479-4486 Request permission

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.

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