On Chen's biharmonic conjecture for hypersurfaces in [formula omitted

Abstract

A longstanding conjecture on biharmonic submanifolds, proposed by Chen in 1991, is that any biharmonic submanifold in a Euclidean space is minimal. In the case of a hypersurface Mn in Rn+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 Mn remains open for n>3. In this paper, we settle Chen's conjecture for hypersurfaces in R5 for n=4.