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On Willmore surfaces in $ S^n$ of flat normal bundle


Author: Peng Wang
Journal: Proc. Amer. Math. Soc. 141 (2013), 3245-3255
MSC (2010): Primary 53A30, 53A07, 53B30
Published electronically: May 16, 2013
MathSciNet review: 3068977
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Abstract: We discuss several kinds of Willmore surfaces of flat normal bundle in this paper. First we show that every S-Willmore surface with flat normal bundle in $ S^n$ must be located in some $ S^3\subset S^n$, from which we characterize the Clifford torus as the only non-equatorial homogeneous minimal surface in $ S^n$ with flat normal bundle, which improves a result of K. Yang. Then we derive that every Willmore two sphere with flat normal bundle in $ S^n$ is conformal to a minimal surface with embedded planer ends in $ \mathbb{R}^3$. We also point out that for a class of Willmore tori, they have a flat normal bundle if and only if they are located in some $ S^3$. In the end, we show that a Willmore surface with flat normal bundle must locate in some $ S^6$.


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Additional Information

Peng Wang
Affiliation: Department of Mathematics, Tongji University, Siping Road 1239, Shanghai, 200092, People’s Republic of China
Email: netwangpeng@tongji.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2013-11683-7
Keywords: Willmore surfaces, S-Willmore surfaces, Willmore sphere, Clifford torus, flat normal bundle
Received by editor(s): November 22, 2011
Published electronically: May 16, 2013
Additional Notes: This work was supported by the Program for Young Excellent Talents in Tongji University, the Tianyuan Foundation of China, grant 10926112, and Project 10901006 of NSFC
Communicated by: Lei Ni
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.