Convex ancient solutions to anisotropic curve shortening flow
Recall that the anisotropic curve shortening flow is a generalization of curve shortening flow, introduced by \textit{J. E. Taylor} [in: Differential geometry. A symposium in honour of Manfredo do Carmo, Proc. Int. Conf., Rio de Janeiro/Bras. 1988, Pitman Monogr. Surv. Pure Appl. Math. 52, 321--336 (1991; Zbl 0725.53011)] and \textit{S. Angenent} and \textit{M. E. Gurtin} [Arch. Ration. Mech. Anal. 108, No. 4, 323--391 (1989; Zbl 0723.73017)] as a physical model for certain crystal interfaces.\N\NIn this interesting paper, a translating solution to anisotropic curve shortening flow is constructed and it is demonstrated that for a given anisotropic factor \N\[\Ng: S^1 \rightarrow \mathbb{R^+},\N\]\Nand a given direction and speed, this translator is unique.\N\NThen an ancient compact solution to anisotropic curve shortening flow is constructed and is shown to be the unique solution, for given \(v,w\), that lies within a slab parallel to \(v\) of width \(w\) and in no smaller slab. Further it is shown that there exist two, up to translation, translating solutions that lie\Nwithin a slab parallel to \(v\) of width \(w\) (and in no smaller slab), one that travels in the \(v\) direction, and one that travels in the \(-v\) direction.
scientific article; zbMATH DE number 7958547