Huisken's Distance Comparison Principle in Higher Codimension

We establish a variant of Huisken’s distance comparison principle for reflection symmetric immersed Curve Shortening flow in ${\mathbb{R}}^n,n\ge 2$. As an application, we show that certain symmetric Curve Shortening flow with a one-to-one convex projection develops a Type~I singularity and becomes asymptotically circular.