Hypersurfaces of \(\mathbb{E}^4\) with harmonic mean curvature vector.

*(English)*Zbl 0944.53005Let \(x:M^n\rightarrow E^m\) be an isometric immersion into Euclidean space. According to the well known Beltrami’s formula, we have \(\Delta x = H\), where \(\Delta\) is the Laplacian of \((M,x)\) and \(H\) its mean curvature vector, so that Euclidean minimal submanifolds are characterized as those with harmonic position vector. B.-Y. Chen [Soochow J. Math. 17, 169-188 (1991; Zbl 0749.53037)]posed the problem of classifying Euclidean submanifolds with harmonic mean curvature vector, that is, those submanifolds satisfying \(\Delta^2x=0\), which are therefore called biharmonic submanifolds. He also conjectured that the only biharmonic Euclidean submanifolds are the minimal ones. The conjecture is known to be true in some special cases: surfaces in \(E^3\) (B.-Y. Chen), curves and submanifolds with constant mean curvature (I. Dimitric), and hypersurfaces in \(E^4\) (T. Hasanis and T. Vlachos), for instance.

In this paper, the author gives a new proof of the Hasanis-Vlachos result (who gave the result in a more ample context) by using a method which is coordinate independent and does not appeal to the use of the computer, in the hope that it can be easier generalized to higher dimensional hypersurfaces.

In this paper, the author gives a new proof of the Hasanis-Vlachos result (who gave the result in a more ample context) by using a method which is coordinate independent and does not appeal to the use of the computer, in the hope that it can be easier generalized to higher dimensional hypersurfaces.

Reviewer: O.J.Garay (Bilbao)

##### MSC:

53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |

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##### References:

[1] | Chen, Total Mean Curvature and Submanifolds of Finite Type (1984) · Zbl 0537.53049 · doi:10.1142/0065 |

[2] | Chen, Some Open Problems and Conjectures on Submanifolds of Finite Type, Soochow J. Math. 17 pp 169– (1991) · Zbl 0749.53037 |

[3] | Chen, Submanifolds of Finite Type and Applications, Proc. Geometry and Topology Research Center, Taegu 3 pp 1– (1993) |

[4] | Chen, Biharmonic Surfaces in Pseudo - Euclidean Spaces, Memoirs of the Fac. of Science, Kyushu University, Series A 45 pp 323– (1991) |

[5] | Dimitri , I. 1989 |

[6] | Dimitri, Submanifolds of Em with Harmonic Mean Curvature Vector, Bull. Inst. Math. Acad. Sinica 20 pp 53– (1992) · Zbl 0778.53046 |

[7] | Garay , O. J. Verstraelen , L. |

[8] | Hasanis, Hypereurfaces in E4 with Harmonic Mean Curvature Vector Field, Math. Nachr. 172 pp 145– (1995) |

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