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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



and the Palais-Smale condition

Author: Anders Linnér
Journal: Trans. Amer. Math. Soc. 350 (1998), 3743-3765
MSC (1991): Primary 58F25; Secondary 58E10, 53C21
MathSciNet review: 1432203
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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 $g$.

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Anders Linnér
Affiliation: Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115

Keywords: Curve-straightening, Palais-Smale condition, gradient trajectories
Received by editor(s): July 17, 1995
Received by editor(s) in revised form: November 5, 1996
Article copyright: © Copyright 1998 American Mathematical Society