Curve-straightening and the Palais-Smale condition

Abstract:

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 $g$.