On Chen's biharmonic conjecture for hypersurfaces in ${\mathbb{R}}^5$

Abstract

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 $n=4$.

23 pages