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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Lower volume estimates and Sobolev inequalities


Authors: Stefano Pigola and Giona Veronelli
Journal: Proc. Amer. Math. Soc. 138 (2010), 4479-4486
MSC (2010): Primary 53C21; Secondary 46E35
Published electronically: July 22, 2010
MathSciNet review: 2680072
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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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Stefano Pigola
Affiliation: Dipartimento di Fisica e Matematica, Università dell’Insubria - Como, via Valleggio 11, I-22100 Como, Italy
Email: stefano.pigola@uninsubria.it

Giona Veronelli
Affiliation: Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, I-20133 Milano, Italy
Email: giona.veronelli@unimi.it

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10514-2
PII: S 0002-9939(2010)10514-2
Received by editor(s): March 12, 2010
Published electronically: July 22, 2010
Communicated by: Michael Wolf
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.