Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean 5-space (Q391160)

A submanifold \(M\) with isometric immersion \(x: M \to\mathbb E^n\) is called biharmonic if \(\Delta^2x=0\) for a rough Laplacian \(\Delta\). In 1991, \textit{B.-Y. Chen} conjectured that all biharmonic submanifolds of Euclidean spaces are minimal [Soochow J. Math 17, No. 2, 169--188 (1991; Zbl 0749.53037)]. While this conjecture in its general form remains open, substantial progress has been made under additional assumptions. It holds, for example, for hypersurfaces in dimension four [\textit{T. Hasanis} and \textit{T. Vlachos}, Math. Nachr. 172, 145--169 (1995; Zbl 0839.53007)]. NEWLINENEWLINENEWLINEIn the present contribution, the author proves Chen's conjecture for biharmonic hypersurfaces in Euclidean spaces of dimension five under the assumption of exactly three distinct principal curvatures. In a remark, he announces a proof for biharmonic hypersurfaces with three distinct principal curvatures in dimension \(n\).