CURVE-STRAIGHTENING AND THE PALAIS-SMALE CONDITION

This paper considers the negative gradient trajectories associated wit h the modified total squared curvature functional integral k(2) + nu d s. The focus is on the limiting behavior as nu tends to zero from the positive side. It is shown that when nu = 0 spaces of curves exist in which some trajectories converge and others diverge. In one instance t he collection of critical points splits into two subsets. As nu tends to zero the critical curves in the first subset tend to the critical p oints present when nu = 0. Meanwhile, all the critical points in the s econd subset have lengths that tend to infinity. It is shown that this is the only way the Palais-Smale condition fails in the present conte xt. The behavior of the second class of critical points supports the v iew that some of the trajectories are 'dragged' all the way to 'infini ty'. When the curves are rescaled to have constant length the Euler fi gure eight emerges as a 'critical point at infinity'. It is discovered that a reflectional symmetry need not be preserved along the trajecto ries. There are examples where the length of the curves along the same trajectory is not a monotone function of the flow-time. It is shown h ow to determine the elliptic modulus of the critical curves in all the standard cases. The modulus p must satisfy 2E(p)/K(p) = 1 +/- \g\/(L) over tilde when the space is limited to curves of fixed length L and the endpoints are separated by the vector g.