论文标题
在陈的Biharmonic猜想中，$ \ mathbb r^5 $
On Chen's biharmonic conjecture for hypersurfaces in $\mathbb R^5$
论文作者
论文摘要
Chen于1991年提出的关于Biharmonic Submanifolds的长期猜想是，{\ IT在欧几里得空间中的任何Biharmonic Submanifold都是最小的}。对于$ \ mathbb r^{n+1} $的高表面$ m^n $，陈的猜想是在1987年左右独立于1987年左右的$ n = 2 $的情况下解决的。 Hasanis和Vlachos在1995年解决了Chen的猜想，以$ n = 3 $的形式进行超表面。但是，Chen将军在Hypersurface $ m^n $上的猜想仍以$ n> 3 $开放。在本文中，我们在$ \ mathbb r^{5} $中质量陈的猜想，$ n = 4 $。
A longstanding conjecture on biharmonic submanifolds, proposed by Chen in 1991, is that {\it any biharmonic submanifold in a Euclidean space is minimal}. In the case of a hypersurface $M^n$ in $\mathbb R^{n+1}$, Chen's conjecture was settled in the case of $n=2$ by Chen and Jiang around 1987 independently. Hasanis and Vlachos in 1995 settled Chen's conjecture for a hypersurface with $n=3$. However, the general Chen's conjecture on a hypersurface $M^n$ remains open for $n> 3$. In this paper, we settle Chen's conjecture for hypersurfaces in $\mathbb R^{5}$ for $n=4$.
