A conjecture proposed by B. Y. Chen states that the only biharmonic submanifolds in Euclidean space are the minimal submanifolds. The theme of this paper is to obtain some partial results on this conjecture under an extra assumption that the submanifold is a \(\delta(r)\)-ideal hypersurface.
A submanifold \(M^n\) in the Euclidean space is biharmonic if its mean curvature vector \(\vec{H}\) satisfies \(\Delta\vec{H}=0\). A \(\delta(r)\)-ideal hypersurface in the Euclidean space is defined via the concept of \(\delta\)-invariants and has a simpler form on its shape operator \(\mathcal{A}\). The main result of this paper is that every \(\delta(r)\)-ideal oriented biharmonic hypersurface with at most \(r+1\) distinct principal curvatures in the Euclidean space \(\mathbb{E}^{n+1}\), \(n\geq 3\), is a minimal hypersurface, where \(r=2,\dots,n-1\). This generalizes an earlier result by B.-Y. Chen and M. I. Munteanu [Differ. Geom. Appl. 31, No. 1, 1–16 (2013; Zbl 1260.53017)] for the cases \(r=2\) and \(r=3\).
Along the proof, the authors also obtain a similar result on \(\delta(r)\)-ideal biconservative hypersurfaces. Here, the biconservativity condition means that \(2\mathcal{A}(\mathrm{grad}~H)+nH\mathrm{grad}~H=0\) which is one of the two equations in the condition of biharmonicity. This means that a biharmonic hypersurface must also be biconservative. In this case, the authors show that every \(\delta(r)\)-ideal oriented biconservative hypersurface with at most \(r+1\) distinct principal curvatures in the Euclidean space \(\mathbb{E}^{n+1}\) \((n\geq 3)\) has constant mean curvature.