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On bi-harmonic hypersurfaces in Euclidean space of arbitrary dimension. (English) Zbl 1323.53065

Summary: 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.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C40 Global submanifolds
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
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[1] DOI: 10.1002/mana.19951720112 · Zbl 0839.53007 · doi:10.1002/mana.19951720112
[2] Dimitric, Bull. Inst. Math. Acad. Sin. 20 pp 53– (1992)
[3] DOI: 10.1002/mana.19981960104 · Zbl 0944.53005 · doi:10.1002/mana.19981960104
[4] DOI: 10.1016/j.difgeo.2012.10.008 · Zbl 1260.53017 · doi:10.1016/j.difgeo.2012.10.008
[5] DOI: 10.1016/j.geomphys.2013.09.004 · Zbl 1283.53005 · doi:10.1016/j.geomphys.2013.09.004
[6] Chen, Proc. Geometry and Topology Research Center, Taegu 3 pp 1– (1993)
[7] Chen, Soochow J. Math 17 pp 169– (1991)
[8] DOI: 10.1142/0065 · doi:10.1142/0065
[9] Chen, Soochow J. Math. 22 pp 117– (1996)
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