Biharmonic hypersurfaces with three distinct principal curvatures in spheres

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 Balmu\c{s}-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 ${\mathbb{S}}^n(1\mathrm{/}\sqrt{2})$ or the Clifford hypersurface $\mathbb S^{n_1}(1/\sqrt2)\times\mathbb S^{n_2}(1/\sqrt2)$with$n_1+n_2=n$and$n_1\neq n_2$. Moreover, we also show that there does not exist proper biharmonic hypersurface with at most three distinct principal curvatures in hyperbolic spaces ${\mathbb{H}}^{n+1}$.