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Hypersurfaces of 𝔼 4 with harmonic mean curvature vector. (English) Zbl 0944.53005

Let x:M n E m be an isometric immersion into Euclidean space. According to the well known Beltrami’s formula, we have Δx=H, where Δ 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 Δ 2 x=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.

MSC:
53A07Higher-dimensional and -codimensional surfaces in Euclidean n-space
53A10Minimal surfaces, surfaces with prescribed mean curvature