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

 

On the weighted forward reduced volume of Ricci flow


Authors: Liang Cheng and Anqiang Zhu
Journal: Proc. Amer. Math. Soc. 141 (2013), 2859-2868
MSC (2010): Primary 53C44
Published electronically: April 9, 2013
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Abstract: In this paper, we introduce the weighted forward reduced volume of Ricci flow. The weighted forward reduced volume, which is related to expanders of Ricci flow, is well-defined on noncompact manifolds and monotone nonincreasing under Ricci flow. Moreover, we show that, just the same as Perelman's reduced volume, the weighted reduced volume entropy has the value $ (4\pi )^{\frac {n}{2}}$ if and only if the Ricci flow is the trivial flow on flat Euclidean space.


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Additional Information

Liang Cheng
Affiliation: School of Mathematics and Statistics, Huazhong Normal University, Wuhan, 430079, People’s Republic of China
Email: math.chengliang@gmail.com

Anqiang Zhu
Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China
Email: anqiangzhu@yahoo.com.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11687-4
PII: S 0002-9939(2013)11687-4
Keywords: Ricci flow, weighted forward reduced volume, Type III singularities, gradient expanding soliton
Received by editor(s): October 31, 2011
Published electronically: April 9, 2013
Additional Notes: The first author was supported by NSF grant of China No. 11171126 and self-determined research funds of CCNU from the colleges basic research and operation of MOE CCNU11A01027
The second author was supported by NSF grant of China No. 11126190
Communicated by: Lei Ni
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.