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 ${\mathbb{R}}^2$, $\alpha \in (\frac{1}{2},1)$ 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 $\alpha \in (\frac{2}{3},1)$. Moreover, we show that any non-compact convex embedded ancient solution of the ${\kappa }^{\alpha }$ flow in ${\mathbb{R}}^2$, $\alpha \in (\frac{1}{2},1)$ must be a translating solution.

Removed some unnecessary complications in the non-compact case and added details to some proofs