A variational problem for submanifolds in a sphere |
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Authors: | Zhen Guo Haizhong Li |
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Institution: | (1) Yunnan Normal University, Kunming, P.R. China;(2) Tsinghua University, Beijing, P.R. China |
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Abstract: | Let
be an n-dimensional submanifold in an (n + p)-dimensional unit sphere S
n + p
, M is called a Willmore submanifold (see 11], 16]) if it is a critical submanifold to the Willmore functional
, where
is the square of the length of the second fundamental form, H is the mean curvature of M. In 11], the second author proved an integral inequality of Simons’ type for n-dimensional compact Willmore submanifolds in S
n + p
. In this paper, we discover that a similar integral inequality of Simons’ type still holds for the critical submanifolds
of the functional
. Moreover, it has the advantage that the corresponding Euler-Lagrange equation is simpler than the Willmore equation. |
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Keywords: | 2000 Mathematics Subject Classification: 53C42 53A10 |
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