Biharmonic $\delta(\lowercase{r})$-ideal hypersurfaces in Euclidean spaces are minimal

Abstract

A submanifold $M^n$ of a Euclidean space $\mathbb{E}^N$ is called biharmonic if $\Delta\vec{H}=0$, where $\vec{H}$ is the mean curvature vector of $M^n$. A well known conjecture of B.Y. Chen states that the only biharmonic submanifolds of Euclidean spaces are the minimal ones. Ideal submanifolds were introduced by Chen as those which receive the least possible tension at each point. In this paper we prove that every $\delta(r)$-ideal biharmonic hypersurfaces in the Euclidean space $\mathbb{E}^{n+1}$ ($n\geq 3$) is minimal. In this way we generalize a recent result of B. Y. Chen and M. I. Munteanu. In particular, we show that every $\delta(r)$-ideal biconservative hypersurface in Euclidean space $\mathbb{E}^{n+1}$ for $n\geq 3$ must be of constant mean curvature.