A distance comparison principle for evolving curves
The author investigates the quotient of extrinsic distance \(d=|\gamma(p)-\gamma(q)|\) and intrinsic distance \(l=\int_p^qds\) for points \(p, q\) on an embedded curve \(\gamma\) in the plane evolving by the curve shortening flow. In the case of closed curves, \(l\) is replaced by \((L/\pi)\sin(l\pi/L)\) where \(L\) is the length of \(\gamma\). The main result is that the ratio \(d/l\), considered as a measure for the straightness of an embedded curve, improves at a local minimum. As a direct application of this result, the author gives an elegant proof for Grayson's convergence result [\textit{M. Grayson}, J. Differ. Geom. 26, 285-314 (1987; Zbl 0667.53001)] where he makes use of the well known fact that singularities in the curve shortening flow equal, after an appropriate rescaling, a selfsimilar evolving curve or a translating soliton, the ``grim reaper curve'' given by \(y=t-\log\cos x\). The latter cannot appear since the ratio between extrinsic and intrinsic distance becomes arbitrarily small, contradicting the main theorem in this paper. Now Grayson's result follows since the only embedded selfsimilar curve is the round circle. In the remainder, some applications to nonclosed curves with boundaries are given.
scientific article; zbMATH DE number 1240915
scientific article