LAUSR

We study biharmonic hypersurfaces in the space forms $$\overline{M}^{6}(c)$$ with at most four distinct principal curvatures and whose second fundamental form is of constant norm. We prove that every such biharmonic hypersurface in $$\overline{M}^{6}(c)$$ has constant mean curvature and constant scalar curvature. In particular, every such biharmonic hypersurface in $$\mathbb {S}^{6}(1)$$ has constant mean curvature and constant scalar curvature. Every such biharmonic hypersurface in Euclidean space $$E^6$$ and in hyperbolic space $$\mathbb {H}^{6}$$ must be minimal and have constant scalar curvature.