Critical curves for the total normal curvature in surfaces of 3-dimensional space forms

A variational problem closely related to the bending energy of curves contained in surfaces of real 3-dimensional space forms is considered. We seek curves in a surface which are critical for the total normal curvature energy (and its generalizations). We start by deriving the first variation formula and the corresponding Euler-Lagrange equations of these energies and apply them to study critical special curves (geodesics, asymptotic lines, lines of curvature) on surfaces. Then, we show that a rotation surface in a real space form for which every parallel is a critical curve must be a special type of a linear Weingarten surface. Finally, we give some classification and existence results for this family of rotation surfaces. (C) 2011 Elsevier Inc. All rights reserved.