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 curvatures ([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 ${\mathbb{E}}^{p+q}$.

18 pages,to appear in Tohoku Math. J

References in corpus (1)

• Properties of biharmonic submanifolds in spheres

Cited by in corpus (5)

• Reduction methods for the bienergy
• H-hypersurfaces with at most 3 distinct principal curvatures in the Euclidean spaces
• Biharmonic hypersurfaces with three distinct principal curvatures in spheres
• Biconservative quasi-minimal immersions into semi-Euclidean spaces
• Chen's conjecture on biharmonic submanifolds in Riemannian manifolds