Lower volume estimates and Sobolev inequalities

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.