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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
Posted:
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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- G. Carron, Inégalités isopérimétriques et inégalités de Faber-Krahn. Séminaire de Théorie Spectrale et Géométrie, No. 13, Année 1994-1995, 63-66. MR 1715957
- 3.
- I. Chavel, Riemannian Geometry: A modern introduction. Cambridge Studies in Mathematics 98. Cambridge University Press, 2006. MR 2229062 (2006m:53002)
- 4.
- J. Cheeger, T.H. Colding, On the structure of spaces with Ricci curvature bounded below. I. J. Differential Geom. 46 (1997), no. 3, 406-480. MR 1484888 (98k:53044)
- 5.
- E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. CIMS Lecture Notes (1999). Courant Institute of Mathematical Sciences. MR 1688256 (2000e:58011)
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- L. Karp, On Stokes' theorem for noncompact manifolds. Proc. Amer. Math. Soc. 82 (1981), 487-490. MR 612746 (83g:58002)
- 7.
- M. Ledoux, On manifolds with non-negative Ricci curvature and Sobolev inequalities. Comm. Anal. Geom. 7 (1999), 347-353. MR 1685586 (2000c:53043)
- 8.
- M. Ledoux, The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), 305-366. MR 1813804 (2002a:58045)
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- V. Gol'dshtein, M. Troyanov, Sobolev inequalities for differential forms and
 -cohomology. Jour. Geom. Anal. 16 (2006), 597-631. MR 2271946 (2008a:58024)
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- S. Pigola, A.G. Setti, M. Troyanov. The topology at infinity of a manifold supporting an -Sobolev inequality. Preprint.
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- S. Pigola, G. Veronelli, An alternative proof of a rigidity theorem for the sharp Sobolev constant. arXiv:1002.0756v1
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- C. Xia, Complete manifolds with nonnegative Ricci curvature and almost best Sobolev constant. Illinois J. Math. 45 (2001), 1253-1259. MR 1894894 (2003h:53056)
- 14.
- S.-H. Zhu, A volume comparison theorem for manifolds with asymptotically nonnegative curvature and its applications. Amer. J. Math. 116 (1994), 669-682. MR 1277451 (95c:53049)
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Additional Information
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
Posted:
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.
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