# Slow and fast forming singularities of the mean curvature flow

Let $$M \times [0, T) \to \mathbb{R}^{n+1}$$ be a mean curvature flow and let $$T$$ be a singular time. Let $$A$$ denote the second fundamental form.

We have a type I singularity if $$\max_{p \in M} |A(p, t)| \le \frac{C}{\sqrt{2(T-t)}}$$ for some constant $$C > 0$$ and a type II singularity otherwise.

Why are type I singularities of the mean curvature flow often called "fast forming singularities" and type II singularities "slow forming"? I don't understand the reason behind this terminology, since the second fundamental form is blowing up slower at type I singularities.

• Can you give examples of this terminology in use in a paper? – T_M Oct 15 '18 at 20:07

I could be wrong but I believe the answer is disappointingly uninteresting: Set $$f(t)\doteqdot \sup_{M\times[0,t]}\vert A\vert^2$$. If the flow is of type I then the remaining time is bounded by $$1/f(t)$$, so a 'singularity' occurs sooner than if the flow is not of type I.