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

Langer, Joel; Singer, David A.

Ann. Global Anal. Geom. 5, No. 2, 133-150 (1987).

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

Keywords:

wave like elastica; bending energy; immersed curve; elastic curves; flow of steepest descent; Palais-Smale condition

Citations:

Zbl 0561.53004

References:

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