Curve-straightening in Riemannian manifolds (Q1107849)

This paper deals with the bending energy of an immersed curve. In a previous paper [Topology 24, 75-88 (1985; Zbl 0561.53004)] the authors studied elastic curves as critical points of the functional \(\int (k\quad 2+\lambda)ds\) where k is the curvature and \(\lambda\) is a real number. Here they study the possibility of a ``flow of steepest descent'' for this functional including a so-called ``strengthened'' Palais-Smale condition. The particular case of curves in the standard two-sphere is examined in detail, and for certain values of \(\lambda\) the stability of the critical points is analyzed. Here the values \(\lambda =8/7\) and \(\lambda =10/9\) for the sphere of radius 1 play a particular role. From the paper: ``The transition between behavior for small values of \(\lambda\) and large ones is somewhat mysterious at present.''