1547671870-The_Ricci_Flow__Chow

6. THE DEGENERATE NECKPINCH

In Theorem 6.45, we shall prove that T < oo only if

lim K (t) = oo.

t / T

65

Because the metric g scales like (length)^2 and Rm = (Rfjk) is dimension-

less, the quantity IRml scales like (length)-^2. We will see in Section 3 of
Chapter 3 that the Ricci flow is equivalent to a parabolic PDE. Since para-
bolic PDE equate time with (length)^2 , the quantity (T - t) K (t) is naturally
dimensionless.
We shall discuss short-time and long-time existence theory for the Ricci
flow in Chapters 3 and 7, respectively. (See also Sections 7 and 8 of Chapter
6.) In Corollary 7. 7, we will prove a short-time existence result which implies
that

(2.68)

c

T-t?. K(t)'

where c > 0 depends only on the dimension n. In other words, the natural

quantity (T - t) K (t) is always bounded from below. One classifies finite-
time singularities, therefore, by whether or not it is bounded from above.
One says the singularity at time T < oo is rapidly forming if the
estimate (2.68) is sharp in the sense that there exists C E [c, oo) such that
for all t < T one has

c c

• 
  
  
  
  • <T-t< --
    K(t) - -K(t)"
  
  
  
  

A rapidly forming singularity is evidently Type I, because (T - t) K (t) :::; C.

On the other hand, if for every C E [c, oo) there exists a time tc < T such

that

(2.69)

c

T - tc > K ( tc) ,

one says the singularity is slowly forming. Equation (2.69) reveals why

this terminology makes sense: there is more time remaining until extinction

than the maximum curvature predicts.

The conjectural picture sketched earlier in this section is very useful

for developing intuition for understanding slowly-forming singularities. Re-

gardless of their role in Hamilton's program for obtaining a geometric and

topological classification of 3-manifolds via the Ricci flow, degenerate neck-

pinches and indeed any slowly forming singularities are interesting in their

own right. Indeed, a rigorous proof of the existence of degenerate neck-

pinches for the Ricci flow and an asymptotic analysis of their behaviors

would be highly valuable in forming productive conjectures concerning cer-

tain analytic (as opposed to topological) properties of singularity formation.