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ON BI-HARMONIC HYPERSURFACES IN EUCLIDEAN SPACE OF ARBITRARY DIMENSION

Published online by Cambridge University Press:  18 December 2014

RAM SHANKAR GUPTA*
Affiliation:
University School of Basic and Applied Sciences, Guru Gobind Singh Indraprastha University, Sector-16C, Dwarka, New Delhi-110078, India E-mail: [email protected]
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Abstract

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The following Chen's bi-harmonic conjecture made in 1991 is well-known and stays open: The only bi-harmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we prove that the bi-harmonic conjecture is true for bi-harmonic hypersurfaces with three distinct principal curvatures of a Euclidean space of arbitrary dimension.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Chen, B. Y., Total mean curvature and submanifolds of finite type (World Scientific, Singapore, 1984).CrossRefGoogle Scholar
2.Chen, B. Y., Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (1991), 169188.Google Scholar
3.Chen, B. Y., Submanifolds of finite type and applications, in Proc. Geometry and Topology Research Center, Taegu, vol. 3 (1993), 148.Google Scholar
4.Chen, B. Y., A report on submanifolds of finite type Soochow J. Math. 22 (1996), 117337.Google Scholar
5.Chen, B. Y. and Munteanu, M. I., Biharmonic ideal hypersurfaces in Euclidean spaces Differ. Geom. Appl. 31 (2013), 116.CrossRefGoogle Scholar
6.Defever, F., Hypersurfaces of E 4 with harmonic mean curvature vector Math. Nachr. 196 (1998), 6169.Google Scholar
7.Dimitric, I., Quadratic representation and submanifolds of finite type, Doctoral Thesis (Michigan State University, 1989).Google Scholar
8.Dimitric, I., Submanifolds of Em with harmonic mean curvature vector Bull. Inst. Math. Acad. Sin. 20 (1992), 5365.Google Scholar
9.Hasanis, Th. and Vlachos, Th., Hypersurfaces in E 4 with harmonic mean curvature vector field, Math. Nachr. 172 (1995), 145169.CrossRefGoogle Scholar
10.Fu, Yu., Biharmonic hypersurfaces with three distinct principal curvatures in the Euclidean 5-space, J. Geom. Phys. 75 (2014), 113119.CrossRefGoogle Scholar