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Hypersurfaces of $\bbfE^4$ 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. {\it 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.

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