Curve-straightening

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper considers the negative gradient trajectories associated with the modified total squared curvature functional . The focus is on the limiting behavior as tends to zero from the positive side. It is shown that when 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 tends to zero the critical curves in the first subset tend to the critical points present when . 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 must satisfy when the space is limited to curves of fixed length and the endpoints are separated by the vector .

**[1]**John Franks,*Geodesics on 𝑆² and periodic points of annulus homeomorphisms*, Invent. Math.**108**(1992), no. 2, 403–418. MR**1161099**, 10.1007/BF02100612**[2]**Wilhelm Klingenberg,*Lectures on closed geodesics*, Springer-Verlag, Berlin-New York, 1978. Grundlehren der Mathematischen Wissenschaften, Vol. 230. MR**0478069****[3]**Joel Langer and David A. Singer,*The total squared curvature of closed curves*, J. Differential Geom.**20**(1984), no. 1, 1–22. MR**772124****[4]**Joel Langer and David A. Singer,*Curve straightening and a minimax argument for closed elastic curves*, Topology**24**(1985), no. 1, 75–88. MR**790677**, 10.1016/0040-9383(85)90046-1**[5]**Joel Langer and David A. Singer,*Curve-straightening in Riemannian manifolds*, Ann. Global Anal. Geom.**5**(1987), no. 2, 133–150. MR**944778**, 10.1007/BF00127856**[6]**Anders Linnér,*Some properties of the curve straightening flow in the plane*, Trans. Amer. Math. Soc.**314**(1989), no. 2, 605–618. MR**989580**, 10.1090/S0002-9947-1989-0989580-5**[7]**Anders Linnér,*Steepest descent as a tool to find critical points of ∫𝑘² defined on curves in the plane with arbitrary types of boundary conditions*, Geometric analysis and computer graphics (Berkeley, CA, 1988) Math. Sci. Res. Inst. Publ., vol. 17, Springer, New York, 1991, pp. 127–138. MR**1081334**, 10.1007/978-1-4613-9711-3_14**[8]**Anders Linnér,*Existence of free nonclosed Euler-Bernoulli elastica*, Nonlinear Anal.**21**(1993), no. 8, 575–593. MR**1245863**, 10.1016/0362-546X(93)90002-A**[9]**Anders Linnér,*Unified representations of nonlinear splines*, J. Approx. Theory**84**(1996), no. 3, 315–350. MR**1377607**, 10.1006/jath.1996.0022**[10]**Richard S. Palais,*Morse theory on Hilbert manifolds*, Topology**2**(1963), 299–340. MR**0158410****[11]**Richard S. Palais and Chuu-Lian Terng,*Critical point theory and submanifold geometry*, Lecture Notes in Mathematics, vol. 1353, Springer-Verlag, Berlin, 1988. MR**972503****[12]**Steinberg, D. H.,*Thesis: Elastic curves in hyperbolic space*, Case Western Reserve University (1995).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
58F25,
58E10,
53C21

Retrieve articles in all journals with MSC (1991): 58F25, 58E10, 53C21

Additional Information

**Anders Linnér**

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

Email:
alinner@math.niu.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-98-01977-1

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