Journal of the Mathematical Society of Japan

Constant mean curvature proper-biharmonic surfaces of constant Gaussian curvature in spheres

Abstract

Constant mean curvature (CMC) surfaces in spheres are investigated under the extra condition of biharmonicity. From the work of Miyata, especially in the flat case, we give a complete description of such immersions and show that for any $h \in (0,1)$ there exist CMC proper-biharmonic planes and cylinders in ${\mathbb S}^5$ with $|H|=h$, while a necessary and sufficient condition on $h$ is found for the existence of CMC proper-biharmonic tori in ${\mathbb S}^5$.

Article information

Source
J. Math. Soc. Japan, Volume 68, Number 3 (2016), 997-1024.

Dates
First available in Project Euclid: 19 July 2016

https://projecteuclid.org/euclid.jmsj/1468956156

Digital Object Identifier
doi:10.2969/jmsj/06830997

Mathematical Reviews number (MathSciNet)
MR3523535

Zentralblatt MATH identifier
06642401

Citation

LOUBEAU, Eric; ONICIUC, Cezar. Constant mean curvature proper-biharmonic surfaces of constant Gaussian curvature in spheres. J. Math. Soc. Japan 68 (2016), no. 3, 997--1024. doi:10.2969/jmsj/06830997. https://projecteuclid.org/euclid.jmsj/1468956156

References

• A. Balmuş, S. Montaldo and C. Oniciuc, Biharmonic PNMC submanifolds in spheres, Ark. Mat., 51 (2013), 197–221.
• A. Balmuş, S. Montaldo and C. Oniciuc, Classification results for biharmonic submanifolds in spheres, Israel J. Math., 168 (2008), 201–220.
• R. Bryant, Minimal surfaces of constant curvature in ${\mathbb S}^n$, Trans. Amer. Math. Soc., 290 (1985), 259–271.
• R. Caddeo, S. Montaldo and C. Oniciuc, Biharmonic submanifolds in spheres, Israel J. Math., 130 (2002), 109–123.
• B.-Y. Chen, A report on submanifolds of finite type, Soochow J. Math., 22 (1996), 117–337.
• B.-Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Mathematics, 1, World Scientific Publishing Co., Singapore, 1984.
• D. Fetcu and A. L. Pinheiro, Biharmonic surfaces with parallel mean curvature in complex space forms, Kyoto J. Math., 55 (2015), 837–855.
• Th. Hasanis and Th. Vlachos, 2-Type surfaces in a hypersphere, Kodai Math. J., 19 (1996), 26–38.
• Th. Hasanis and Th. Vlachos, Spherical 2-type surfaces, Arch. Math. (Basel), 67 (1996), 430–440.
• E. Loubeau and C. Oniciuc, Biharmonic surfaces of constant mean curvature, Pacific J. Maths., 271 (2014), 213–230.
• Y. Miyata, 2-Type surfaces of constant curvature in ${\mathbb S}^n$, Tokyo J. Math., 11 (1988), 157–204.
• C. Oniciuc, Tangency and Harmonicity Properties, PhD Thesis, Geometry Balkan Press 2003, http://www.mathem.pub.ro/dgds/mono/dgdsmono.htm
• 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.
• J. H. Sampson, Some properties and applications of harmonic mappings, Ann. Sci. ENS, 11 (1978), 211–228.
• T. Sasahara, Biharmonic Lagrangian surfaces of constant mean curvature in complex space forms, Glasg. Math. J., 49 (2007), 497–507.
• T. Sasahara, Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors, Publ. Math. Debrecen, 67 (2005), 285–303.
• T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380–385.