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

 

Volume estimate about shrinkers


Authors: Xu Cheng and Detang Zhou
Journal: Proc. Amer. Math. Soc. 141 (2013), 687-696
MSC (2010): Primary 53C20
Published electronically: September 27, 2012
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Abstract: We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finiteness, polynomial volume growth and properness of an immersed self-shrinker in Euclidean space.


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Additional Information

Xu Cheng
Affiliation: Instituto de Matematica, Universidade Federal Fluminense, Niterói, RJ 24020, Brazil
Email: xcheng@impa.br

Detang Zhou
Affiliation: Instituto de Matematica, Universidade Federal Fluminense, Niterói, RJ 24020, Brazil
Email: zhou@impa.br

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11922-7
PII: S 0002-9939(2012)11922-7
Received by editor(s): July 4, 2011
Published electronically: September 27, 2012
Additional Notes: Both authors were partially supported by CNPq and FAPERJ, Brazil.
Communicated by: Chuu-Lian Terng
Article copyright: © Copyright 2012 American Mathematical Society