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.

  • $\begingroup$ Can you give examples of this terminology in use in a paper? $\endgroup$ – 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.


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