I've been reading Huisken's paper on his distance comparison principle and he remarked that in particular his theorem rules out the formation of type II singularities. These are singularities where in particular cusps form. Why is this true? For reference I will include his principle below
Huisken's Distance Comparison Principle
Let $F:\Gamma\times[0,T]\rightarrow\mathbb{R}^2$ be a smooth embedded solution of the curve shortening flow (1.1). Let $\Gamma\neq S^1$, such that $l$ is smoothly defined on $\Gamma\times \Gamma$. Suppose $d/l$ attains a local minimum at (p,q) in the interior of $\gamma\times\gamma$ at time $t_0\in[0,T]$. Then$$\frac{d}{dt}(d/l)(p,q,t_0)\geq0$$ with equality if and only if $\Gamma$ is a straight line.
Note: $$d(p,q,t)=|F(p,t)-F(q,t)|$$ and $$l(p,q,t)=|\int_p^qds_t|$$
Reference: A Distance Comparison Principle for Evolving Curves by Huisken
Huisken's result shows that the ratio between extrinsic and intrinsic distances is increasing, and since $S^1$ is compact, this tells us that the minimum of the ratio at any time slice is increasing. On the other hand, it is well know that the blowup of a type II singularity leads to the grim reaper (cf. Angenent's paper ), but on a grim reaper, this distance ratio has no positive lower bound. Thus, this could not happen for an embedded curve.