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. We study biminimalcurves in 2-dimensional Riemannian man-ifolds of constant curvature. IntroductionElastic curves provide examples of classically known geometric variationalproblem. A plane curve is said to be an elastic curve if it is a critical point ofthe elastic energy, or equivalently a critical point of the total squared curvature[9].In this paper, we study another geometric variational problem of curves inRiemannian 2-manifolds of constant curvature. The Euler-Lagrange equationstudied 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 saidto be biharmonic if it is a critical point of the bienergy functional:E 2 (φ) =Z M |τ(φ)| 2 dv g ,where τ(φ) = tr ∇dφ is the tension eld 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 isnot 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 curveson 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 acritical point of the bienergy functional under all normal variations. Thus thebiminimality is weaker than biharmonicity for isometric immersions, in general.

BIMINIMAL CURVES IN 2-DIMENSIONAL SPACE FORMS