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 )
https://doi.org/10.15407/mag16.02.161

Biharmonic Hopf Hypersurfaces of Complex Euclidean Space and Odd Dimensional Sphere

Najma Mosadegh

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

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

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