Symmetrized curve-straightening.
The curve-straightening flow with \(\gamma\geq 0\) is the negative gradient flow associated with the modified total squared curvature functional: \(\nu\to \int_\gamma{1\over 2} k^2+\gamma \,ds\) on the space \(X\) of curves, the larger possible Sobolev space on which the modified total squared curvature functional is defined with the simplest standard metric. The author's choice facilitates the computation of the gradient and its projection onto tangent spaces in the case of constraints. It also simplifies several of the analytical arguments required when verifying the Palais-Smale condition. In this context, the purpose of this paper is to answer questions concerning the failure of the traditional curve-straightening flow to preserve certain symmetries of reflection. The main result is the proof of the existence of such curve-straightening flows. This investigation will also explain the phenomena of ``spinning'' as observed in several examples, in the traditional flow. The paper concludes with a number of graphical illustrations that show preserved symmetry and removal of spinning.
scientific article; zbMATH DE number 1970794