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Hypersurfaces of $$\mathbb{E}^4$$ with harmonic mean curvature vector. (English) Zbl 0944.53005
Let $$x:M^n\rightarrow E^m$$ 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.
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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