Received January 9, 2019, revised November 28, 2019.

Abstract

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 $n_1 + n_2 =2n$.