## Curve-straightening in Riemannian manifolds.(English)Zbl 0653.53032

This paper deals with the bending energy of an immersed curve. In a previous paper [Topology 24, 75-88 (1985; Zbl 0561.53004)] the authors studied elastic curves as critical points of the functional $$\int (k\quad 2+\lambda)ds$$ where k is the curvature and $$\lambda$$ is a real number. Here they study the possibility of a “flow of steepest descent” for this functional including a so-called “strengthened” Palais-Smale condition. The particular case of curves in the standard two-sphere is examined in detail, and for certain values of $$\lambda$$ the stability of the critical points is analyzed. Here the values $$\lambda =8/7$$ and $$\lambda =10/9$$ for the sphere of radius 1 play a particular role. From the paper: “The transition between behavior for small values of $$\lambda$$ and large ones is somewhat mysterious at present.”
Reviewer: W.Kühnel

### MSC:

 53C40 Global submanifolds 58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable) 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Zbl 0561.53004
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### References:

 [1] U. Abresch: Constant Mean Curvature Tori in Terms of Elliptic Functions, preprint. [2] P. F. Byrd and M. D. Friedman: Handbook of elliptic integrals for engineers and physicists, Springer, Berlin 1954. · Zbl 0055.11905 [3] M. Gage and R. Hamilton: The Heat Equation Shrinking Convex Plane Ccurves, J. Differential Geometry23 (1986), 69-96. · Zbl 0621.53001 [4] J. Langer and D. A. Singer: Total Squared Curvature of Closed Curves, J. Differential Geometry20 (1984), 1-22. · Zbl 0554.53013 [5] J. Langer and D. A. Singer: Curves in the Hyperbolic Plane and Mean Curvature of Tori in 3-Space, Boll. London Math. Soc.16 (1984), 531-534. · Zbl 0554.53014 [6] J. Langer and D. A. Singer: Curve Straightening and a Minimax Argument for Closed Elastic Curves, Topology24 (1985), 75-88. · Zbl 0561.53004 [7] R. Miller: Nonlinear Volterra Equations, W. A. Benjamin, Menlo Park 1971. [8] A. Trnkall: Hopf Tori inR 3, Invent. Math.81 (1986), 379-386. [9] A. Tromba: A General Approach to Morse Theory, J. Differential Geometry12 (1977), 47-85. · Zbl 0344.58012
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