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Relating diameter and mean curvature for Riemannian submanifolds


Authors: Jia-Yong Wu and Yu Zheng
Journal: Proc. Amer. Math. Soc. 139 (2011), 4097-4104
MSC (2010): Primary 53C40, 57R42
DOI: https://doi.org/10.1090/S0002-9939-2011-10848-7
Published electronically: March 25, 2011
MathSciNet review: 2823054
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Abstract | References | Similar Articles | Additional Information

Abstract: Given an $ m$-dimensional closed connected Riemannian manifold $ M$ smoothly isometrically immersed in an $ n$-dimensional Riemannian manifold $ N$, we estimate the diameter of $ M$ in terms of its mean curvature field integral under some geometric restrictions, and therefore generalize a recent work of P. M. Topping in the Euclidean case (Comment. Math. Helv., 83 (2008), 539-546).


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Additional Information

Jia-Yong Wu
Affiliation: Department of Mathematics, Shanghai Maritime University, Haigang Avenue 1550, Shanghai 201306, People’s Republic of China
Email: jywu81@yahoo.com

Yu Zheng
Affiliation: Department of Mathematics, East China Normal University, Dong Chuan Road 500, Shanghai 200241, People’s Republic of China
Email: zhyu@math.ecnu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2011-10848-7
Keywords: Mean curvature, Riemannian submanifolds, geometric inequalities, diameter estimate.
Received by editor(s): January 24, 2010
Received by editor(s) in revised form: September 23, 2010
Published electronically: March 25, 2011
Additional Notes: This work is partially supported by NSFC10871069.
Communicated by: Michael Wolf
Article copyright: © Copyright 2011 American Mathematical Society

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