Ancient solutions for flow by powers of the curvature in $\mathbb R^2$

Abstract

We construct a new compact convex embedded ancient solution of the $\kappa^\alpha$ flow in $\mathbb R^2$, $\alpha\in(\frac12,1)$ that lies between two parallel lines. Using this solution we classify all convex ancient solutions of the $\kappa^\alpha$ flow in $\mathbb R^2$, for $\alpha\in(\frac23,1)$. Moreover, we show that any non-compact convex embedded ancient solution of the $\kappa^\alpha$ flow in $\mathbb R^2$, $\alpha\in(\frac12,1)$ must be a translating solution.