Abstract
First, we classify proper biharmonic Hopf real hypersurfaces in \({\mathbb {C}}P^2\). Next, we classify proper biharmonic real hypersurfaces with two distinct principal curvatures in \({\mathbb {C}}P^n\), where \(n\ge 2\). Finally, we prove that biharmonic ruled real hypersurfaces in \({\mathbb {C}}P^n\) are minimal, where \(n\ge 2\).
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Adachi, T., Bao, T., Maeda, S.: Congruence classes of minimal ruled real hypersurfaces in a nonflat complex space form. Hokkaido Math. J. 43, 137–150 (2014)
Balmuş, A., Montaldo, S., Oniciuc, C.: Classification results for biharmonic submanifolds in spheres. Isr. J. Math. 168, 201–220 (2008)
Fetcu, D., Loubeau, E., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds in \({\mathbb{C}}P^n\). Math. Z. 266, 505–531 (2010)
Ichiyama, T., Inoguchi, J., Urakawa, H.: Biharmonic map and bi-Yang-Mills fields. Note Mat. 28(Suppl. 1), 233–275 (2008)
Ivey, T.A., Ryan, P.J.: Hypersurfaces in \({\mathbb{C}}P^2\) and \({\mathbb{C}}H^2\) with two distinct principal curvatures. Glasg. Math. J. 58, 137–152 (2016)
Jiang, G.Y.: \(2\)-Harmonic maps and their first and second variational formulas. Chin. Ann. Math. A 7, 389–402 (1986) (in Chinese)
Kimura, M.: Real hypersurfaces and complex submanifolds in complex projective space. Trans. Am. Math. Soc. 296, 137–149 (1986)
Kimura, M.: Sectional curvatures of holomorphic planes on a real hypersurface in \(P^n({\mathbb{C}})\). Math. Ann. 276, 487–497 (1987)
Niebergall, R., Ryan, P.J.: Real hypersurfaces in complex space forms. Tight Taut Submanifolds 32, 233–305 (1997)
Takagi, R.: On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10, 495–506 (1973)
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Sasahara, T. Classification Theorems for Biharmonic Real Hypersurfaces in a Complex Projective Space. Results Math 74, 136 (2019). https://doi.org/10.1007/s00025-019-1062-3
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DOI: https://doi.org/10.1007/s00025-019-1062-3