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Variational problems of total mean curvature of submanifolds in a sphere


Authors: Zhen Guo and Bangchao Yin
Journal: Proc. Amer. Math. Soc. 144 (2016), 3563-3568
MSC (2010): Primary 53C17, 53C40, 53C42
Published electronically: February 3, 2016
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Abstract: Let $ \mathbf {H}$ be the mean curvature vector of an $ n$-dimensional submanifold in a Riemannian manifold. The functional $ \mathcal {H}=\int \Vert\mathbf {H}\Vert^{n}$ is called the total mean curvature functional. In this paper, we present the first variational formula of $ \mathcal {H}$ and then, for a critical surface of $ \mathcal {H}$ in the ($ 2+p$)-dimensional unit sphere $ \mathbb{S}^{2+p}$, we establish the relationship between the integral of an extrinsic quantity of the surfaces and its Euler characteristic number.


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Additional Information

Zhen Guo
Affiliation: Department of Mathematics, Yunnan Normal University, Kunming 650500, People’s Republic of China
Email: gzh2001y@yahoo.com

Bangchao Yin
Affiliation: Department of Mathematics, Yunnan Normal University, Kunming 650500, People’s Republic of China
Email: mathyinchao@163.com

DOI: https://doi.org/10.1090/proc/13009
Keywords: Submanifolds, total mean curvature, variation, Euler characteristic
Received by editor(s): July 30, 2015
Received by editor(s) in revised form: October 15, 2015
Published electronically: February 3, 2016
Additional Notes: The authors were supported by project numbers 11161056 and 11531012 of the National Natural Science Foundation of China
Communicated by: Guofang Wei
Article copyright: © Copyright 2016 American Mathematical Society