BIMINIMAL CURVES IN 2-DIMENSIONAL SPACE FORMS

Abstract

Abstract. 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 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 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.