Commun. Korean Math. Soc. 27 (2012), No. 4, pp. 771–780 http://dx.doi.org/10.4134/CKMS.2012.27.4.771 BIMINIMAL CURVES IN 2-DIMENSIONAL SPACE FORMS Jun-ichi Inoguchi and Ji-Eun Lee Abstract. We study biminimal curves in 2-dimensional Riemannian man- ifolds of constant curvature. Introduction Elastic curves provide examples of classically known geometric variational problem. A plane curve is said to be an elastic curve if it is a critical point of the elastic energy, or equivalently a critical point of the total squared curvature [9]. In this paper, we study another geometric variational problem of curves in Riemannian 2-manifolds of constant curvature. The Euler-Lagrange equation studied in this paper is derived from the theory of biharmonic maps in Rie- mannian geometry. A smooth map φ :(M,g) → (N,h) between Riemannian manifolds is said to be biharmonic if it is a critical point of the bienergy functional: E 2 (φ)=  M |τ (φ)| 2 dv g , where τ (φ) = tr ∇dφ is the tension field of φ. Clearly, if φ is harmonic, i.e., τ (φ) = 0, then φ is biharmonic. A biharmonic map is said to be proper if it is not harmonic. Chen and Ishikawa [3] studied biharmonic curves and surfaces in semi- Euclidean space (see also [6]). Caddeo, Montaldo and Piu [1] studied bihar- monic curves on Riemannian 2-manifolds. They showed that biharmonic curves on Riemannian 2-manifolds of non-positive curvature are geodesics. Proper bi- harmonic curves on the unit 2-sphere are small circles of radius 1/ √ 2. Next, Loubeau and Montaldo introduced the notion of biminimal immersion [10]. An isometric immersion φ :(M,g) → (N,h) is said to be biminimal if it is a critical point of the bienergy functional under all normal variations. Thus the biminimality is weaker than biharmonicity for isometric immersions, in general. Received July 21, 2011. 2010 Mathematics Subject Classification. 58E20. Key words and phrases. biminimal curves, elliptic functions. c 2012 The Korean Mathematical Society 771