1547671870-The_Ricci_Flow__Chow

64 2. SPECIAL AND LIMIT SOLUTIONS

FIGURE 6. A cusp forming

On the other hand, the singularity is classified as Type Ila if

sup IRml · (T - t) = oo.

Mnx[O,T)

Type Ila singularities are called 'slowly forming'. (We will explain this

terminology below.) The heuristic picture we have described motivates the

following question about the formation of slowly forming singularities.

PROBLEM 2.47. Given a compact smooth 3-manifold M^3 and a one-

parameter family of initial metrics fas : /3 E [O, 1]} such that the Ricci flow

starting at g 1 forms a Type I singularity model which is a quotient 53 /f1 of

the 3-sphere, while the Ricci flow starting at go forms a Type I singularity

model which is a quotient (5^2 x IR) /fo of the cylinder, does there exist

f3 E (0, 1) such that the flow starting at g 13 forms a Type Ila singularity?

It is expected that at any f3 E (0, 1) where there is a transition in the

singularity model, a Type Ila singularity should form.

6.2. An intuitive picture of slowly forming singularities. Be-

cause a Type Ila singularity has the property that

sup IRml · (T - t) = oo,

Mnx[O,T)

the terminology 'slowly forming' may initially seem counterintuitive. On the

contrary, it is in fact perfectly logical, as we now explain.

Let us first explore why supMnx[O,T) IRml · (T- t) is a meaningful quan-

tity for finite-time singularities. Let (Mn,g (t)) be a solution of the Ricci

flow defined on a maximal time interval a :S: t < T, and define

K (t) ~ sup IRm (x, t)I.

xEMn