Biharmonic Hopf hypersurfaces of complex Euclidean space and odd dimensional sphere

In this paper, biharmonic Hopf hypersurfaces in the complex Euclidean space $C^{n+1}$ and in the odd dimensional sphere $S^{2n+1}$ are considered. We prove that the biharmonic Hopf hypersurfaces in $C^{n+1}$ are minimal. Also, we determine that the Weingarten operator $A$ of a biharmonic pseudo-Hopf hypersurface in the unit sphere $S^{2n+1}$ has exactly two distinct principal curvatures at each point if the gradient of the mean curvature belongs to $D^\perp$, and thus is an open part of the Clifford hypersurface $S^{n_1} (1/\sqrt{2})\times S^{n_2} (1/\sqrt{2})$, where $n_1 + n_2 =2n$.