Biharmonic hypersurfaces with three distinct principal curvatures in spheres

Abstract

We obtain a complete classification of proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces with arbitrary dimension. Precisely, together with known results of Balmuş-Montaldo-Oniciuc, we prove that compact orientable proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces ${\mathbb{S}}^{n+1}$ are either the hypersphere ${\mathbb{S}}^n(1/\sqrt{2})$ or the Clifford hypersurface ${\mathbb{S}}^{n_1}(1/\sqrt{2})\times {\mathbb{S}}^{n_2}(1/\sqrt{2})$ with $n_1+n_2=n$ and $n_1\ne n_2$. Moreover, we also show that there does not exist proper biharmonic hypersurface with at most three distinct principal curvatures in hyperbolic spaces ${\mathbb{H}}^{n+1}$.

14 pages, accepted by Math. Nachr. on 10/24/2014