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

TL;DR

This paper proves that biharmonic $delta(r)$-ideal hypersurfaces in Euclidean spaces are necessarily minimal, extending previous results and confirming the conjecture that biharmonic submanifolds in Euclidean space are minimal.

Contribution

It establishes that all $delta(r)$-ideal biharmonic hypersurfaces in Euclidean spaces are minimal, generalizing recent findings and linking biharmonicity with minimality.

Findings

$delta(r)$-ideal biharmonic hypersurfaces are minimal

$delta(r)$-ideal biconservative hypersurfaces have constant mean curvature

extends previous results on biharmonic submanifolds

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.