We consider an embedded convex ancient solution $\Gamma_t$ to the curve shortening flow in $\mathbb{R}^2$. We prove that there are only two possibilities: the family $\Gamma_t$ is either the family of contracting circles, which is a type I ancient solution, or the family of evolving Angenent ovals, which correspond to a type II ancient solution to the curve shortening flow. We also give a necessary and sufficient curvature condition for an embedded, closed ancient solution to the curve shortening flow to be convex.

中文翻译：

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曲线缩短流的紧凑古解的分类

我们考虑 $\mathbb{R}^2$ 中曲线缩短流的嵌入凸古解 $\Gamma_t$。我们证明只有两种可能性：$\Gamma_t$ 族要么是收缩圆族，这是 I 型古解，要么是演化 Angenent 椭圆族，对应于曲线的 II 型古解缩短流量。我们还给出了曲线缩短流的一个嵌入的、封闭的古解是凸的一个充分必要的曲率条件。