Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean space

Abstract

The well known Chen's conjecture on biharmonic submanifolds states that a biharmonic submanifold in a Euclidean space is a minimal one ([10--13, 16, 18--21, 8]). For the case of hypersurfaces, we know that Chen's conjecture is true for biharmonic surfaces in $\mathbb E^3$ ([10], [24]), biharmonic hypersurfaces in $\mathbb E^4$ ([23]), and biharmonic hypersurfaces in $\mathbb E^m$ with at most two distinct principal curvature ([21]). The most recent work of Chen-Munteanu [18] shows that Chen's conjecture is true for $\delta(2)$-ideal hypersurfaces in $\mathbb E^m$, where a $\delta(2)$-ideal hypersurface is a hypersurface whose principal curvatures take three special values: $\lambda_1, \lambda_2$ and $\lambda_1+\lambda_2$. In this paper, we prove that Chen's conjecture is true for hypersurfaces with three distinct principal curvatures in $\mathbb E^m$ with arbitrary dimension, thus, extend all the above-mentioned results. As an application we also show that Chen's conjecture is true for $O(p)\times O(q)$-invariant hypersurfaces in Euclidean space $\mathbb E^{p+q}$.