Uniqueness of Tangent Flows at Infinity for Finite-Entropy Shortening Curves

Abstract

In this paper, we prove that an ancient smooth curve-shortening flow with finite entropy embedded in \(\mathbb{R}^{2}\) has a unique tangent flow at infinity. To this end, we show that its rescaled flows backwardly converge to a line with multiplicity m≥3 exponentially fast in any compact region, unless the flow is a shrinking circle, a static line, a paper clip, or a translating grim reaper. In addition, we figure out the exact numbers of tips, vertices, and inflection points of the curves at negative enough time. Moreover, the exponential growth rate of graphical radius and the convergence of vertex regions to grim reaper curves will be shown.