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Hypersurfaces in \(E^ 4\) with harmonic mean curvature vector field. (English) Zbl 0839.53007
Given an immersion in the Euclidean space \(x:M \to \mathbb{R}^m\), it is well known that the immersion is harmonic, i.e. \(\Delta x=0\), if and only if \(M\) is minimal. In this paper the author studies biharmonic immersions, i.e. such that \(\Delta^2 x=0\). He proves that \(M\) is a biharmonic hypersurface in \(\mathbb{R}^4\) if and only if \(M\) is a minimal hypersurface.

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
Full Text: DOI
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