1547671870-The_Ricci_Flow__Chow

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  1. THE DEGENERATE NECKPINCH


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


lim K (t) = oo.
t / T

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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.
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