Curve-straightening and the Palais-Smale condition

This paper considers the negative gradient trajectories associated with the modified total squared curvature functional $\int k^{2} +\nu ds$. 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 the collection of critical points splits into two subsets. As $\nu$ tends to zero the critical curves in the first subset tend to the critical points present when $\nu =0$. Meanwhile, all the critical points in the second subset have lengths that tend to infinity. It is shown that this is the only way the Palais-Smale condition fails in the present context. The behavior of the second class of critical points supports the view that some of the trajectories are `dragged' all the way to `infinity'. When the curves are rescaled to have constant length the Euler figure eight emerges as a `critical point at infinity'. It is discovered that a reflectional symmetry need not be preserved along the trajectories. 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 how to determine the elliptic modulus of the critical curves in all the standard cases. The modulus $p$ must satisfy $2E(p)/K(p)=1\pm |g|/\widetilde L$ when the space is limited to curves of fixed length $\widetilde L$ and the endpoints are separated by the vector $g$.