论文标题
Biharmonic $δ（\ lowercase {r}）$ - 欧几里得空间中的理想超曲面很小
Biharmonic $δ(\lowercase{r})$-ideal hypersurfaces in Euclidean spaces are minimal
论文作者
论文摘要
如果$ \Δ\ vec {h} = 0 $，euclidean space $ \ mathbb {e}^n $的submanifold $ m^n $称为biharmonic，其中$ \ vec {h} $是$ m^n $的平均曲率向量。 B.Y.的众所周知的猜想陈指出，欧几里得空间的唯一Biharmonic submanifolds是最小的。理想的子手机是由陈引入的，因为那些在每个点处受到最小张力的张力。在本文中，我们证明了欧几里得空间中的每一个$δ（r）$ - 理想的Biharmonic Hyperfaces $ \ mathbb {e}^{n+1} $（$ n \ geq 3 $）是最小的。通过这种方式，我们概括了B. Y. Chen和M. I. Munteanu的最新结果。特别是，我们表明，每一个$δ（r）$ - 欧几里得空间中的理想双守平性高表情$ \ mathbb {e}^{n+1} $ for $ n \ geq 3 $必须具有恒定的平均值曲率。
A submanifold $M^n$ of a Euclidean space $\mathbb{E}^N$ is called biharmonic if $Δ\vec{H}=0$, 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 $δ(r)$-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 $δ(r)$-ideal biconservative hypersurface in Euclidean space $\mathbb{E}^{n+1}$ for $n\geq 3$ must be of constant mean curvature.
