## Kyoto Journal of Mathematics

### Biharmonic surfaces with parallel mean curvature in complex space forms

#### Abstract

We classify complete biharmonic surfaces with parallel mean curvature vector field and nonnegative Gaussian curvature in complex space forms.

#### Article information

Source
Kyoto J. Math., Volume 55, Number 4 (2015), 837-855.

Dates
Received: 2 July 2014
Revised: 2 October 2014
Accepted: 2 October 2014
First available in Project Euclid: 25 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1448460081

Digital Object Identifier
doi:10.1215/21562261-3157766

Mathematical Reviews number (MathSciNet)
MR3479312

Zentralblatt MATH identifier
1336.53067

#### Citation

Fetcu, Dorel; Pinheiro, Ana Lucia. Biharmonic surfaces with parallel mean curvature in complex space forms. Kyoto J. Math. 55 (2015), no. 4, 837--855. doi:10.1215/21562261-3157766. https://projecteuclid.org/euclid.kjm/1448460081

#### References

• [1] H. Alencar, M. do Carmo, and R. Tribuzy, A Hopf theorem for ambient spaces of dimensions higher than three, J. Differential Geom. 84 (2010), 1–17.
• [2] A. Balmuş, S. Montaldo, and C. Oniciuc, Classification results for biharmonic submanifolds in spheres, Israel J. Math. 168 (2008), 201–220.
• [3] A. Balmuş, S. Montaldo, and C. Oniciuc, Biharmonic PNMC submanifolds in spheres, Ark. Mat. 51 (2013), 197–221.
• [4] A. Balmuş and C. Oniciuc, Biharmonic submanifolds with parallel mean curvature vector field in spheres, J. Math. Anal. Appl. 386 (2012), 619–630.
• [5] M. H. Batista da Silva, Simons type equation in $\mathbb{S}^{2}\times\mathbb{R}$ and $\mathbb{H}^{2}\times\mathbb{R}$ and applications, Ann. Inst. Fourier (Grenoble) 61 (2011), 1299–1322.
• [6] B.-Y. Chen, A report on submanifolds of finite type, Soochow J. Math. 22 (1996), 117–337.
• [7] B.-Y. Chen and K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974), 257–266.
• [8] S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), 195–204.
• [9] J. Eells, Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160.
• [10] J.-H. Eschenburg and R. Tribuzy, Existence and uniqueness of maps into affine homogeneous spaces, Rend. Sem. Mat. Univ. Padova 89 (1993), 11–18.
• [11] D. Fetcu, Surfaces with parallel mean curvature vector in complex space forms, J. Differential Geom. 91 (2012), 215–232.
• [12] D. Fetcu, E. Loubeau, S. Montaldo, and C. Oniciuc, Biharmonic submanifolds of $\mathbb{C}P^{n}$, Math. Z. 266 (2010), 505–531.
• [13] D. Fetcu and C. Oniciuc, Biharmonic integral $\mathcal{C}$-parallel submanifolds in $7$-dimensional Sasakian space forms, Tohoku Math. J. (2) 64 (2012), 195–222.
• [14] D. Fetcu, C. Oniciuc, and H. Rosenberg, Biharmonic submanifolds with parallel mean curvature in $\mathbb{S}^{n}\times\mathbb{R}$, J. Geom. Anal. 23 (2013), 2158–2176.
• [15] A. Huber, On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32 (1957), 13–72.
• [16] G. Y. Jiang, $2$-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7 (1986), 389–402.
• [17] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, I, Interscience, New York, 1963.
• [18] E. Loubeau and C. Oniciuc, Biharmonic surfaces of constant mean curvature, Pacific J. Math. 271 (2014), 213–230.
• [19] S. Maeda and T. Adachi, Holomorphic helices in a complex space form, Proc. Amer. Math. Soc. 125 (1997), 1197–1202.
• [20] S. Maeda and Y. Ohnita, Helical geodesic immersions into complex space forms, Geom. Dedicata 30 (1989), 93–114.
• [21] S. Maeta and H. Urakawa, Biharmonic Lagrangian submanifolds in Kähler manifolds, Glasg. Math. J. 55 (2013), 465–480.
• [22] Y.-L. Ou and Z.-P. Wang, Constant mean curvature and totally umbilical biharmonic surfaces in $3$-dimensional geometries, J. Geom. Phys. 61 (2011), 1845–1853.
• [23] H. Reckziegel, “Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion” in Global Differential Geometry and Global Analysis 1984 (Berlin, 1984), Lecture Notes in Math. 1156, Springer, Berlin, 1985, 264–279.
• [24] T. Sasahara, Biharmonic Lagrangian surfaces of constant mean curvature in complex space forms, Glasg. Math. J. 49 (2007), 497–507.
• [25] N. Sato, Totally real submanifolds of a complex space form with nonzero parallel mean curvature vector, Yokohama Math. J. 44 (1997), 1–4.
• [26] J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105.
• [27] W. Zhang, New examples of biharmonic submanifolds in $\mathbb{C}P^{n}$ and $\mathbb{S}^{2n+1}$, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 57 (2011), 207–218.