Journal of the Mathematical Society of Japan

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

Eric LOUBEAU and Cezar ONICIUC

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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

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1468956156

Digital Object Identifier
doi:10.2969/jmsj/06830997

Mathematical Reviews number (MathSciNet)
MR3523535

Zentralblatt MATH identifier
06642401

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53C43: Differential geometric aspects of harmonic maps [See also 58E20] 58E20: Harmonic maps [See also 53C43], etc.

Keywords
biharmonic map constant mean curvature surfaces

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.


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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.