Journal of Mathematical Physics, Analysis, Geometry 2020, vol. 16, No 2, pp. 161-173   https://doi.org/10.15407/mag16.02.161     ( to contents , go back )

### Biharmonic Hopf Hypersurfaces of Complex Euclidean Space and Odd Dimensional Sphere

Depertment of Mathematics Azarbaijan Shahid Madani University, Tabriz 53751 71379, Iran

Esmaiel Abedi

Depertment of Mathematics Azarbaijan Shahid Madani University, Tabriz 53751 71379, Iran
E-mail: esabedi@azaruniv.ac.ir

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 $n_1 + n_2 =2n$.

Mathematics Subject Classification 2000: 53A10, 53C42
Key words: biharmonic hypersurfaces, Hopf hypersurfaces, Chen's conjecture