Weconsidercompletemanifoldswithasymptoticallynon-negative curvature which enjoy a Euclidean-type Sobolev inequality and we get an ex- plicit 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.
Lower volume estimates and Sobolev inequalities
PIGOLA, STEFANO;
2010-01-01
Abstract
Weconsidercompletemanifoldswithasymptoticallynon-negative curvature which enjoy a Euclidean-type Sobolev inequality and we get an ex- plicit 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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