1547671870-The_Ricci_Flow__Chow

5. THE NECKPINCH 53

log K1 breaks scale invariance. Our bound on F will thus show that Ko/ K 1
becomes small whenever K1 is large, in particular near a forming neckpinch.
The main results obtained in this part of the proof show that the singu-
larity is Type I (rapidly forming) and estimate its asymptotics. (We shall
discuss the classification of singularities in Section 1 of Chapter 8.)

PROPOSITION 2.36. Let g (t) : 0 :::; t < T be a maximal solution of the

Ricci fiow having the form (2.43) such that 1-iPsl :::; 1 and R 2: 0. Assume

that the solution has at least one neck.

(1) A singularity occurs at the smallest neck at some time T < oo. This

singularity is of Type I; in particular, there exists C = C ( n, go)

such that

c

IRml :'.::: T - (

(2) There exists C = C (n, go) such that

K

L [log L + 2 - log Lmin (O)] :'.::: C.

(3) Let s (t) denote the location of the smallest neck. Then there are

constants 8 > 0 and C < oo such that fort sufficiently close to T,

one has the estimate

Vi C (s-s)

2

1<- <1+ --

• rmin - - log rmin rmin
  in the inner layer Is - sl :'.::: 2r min v - log r min, and the estimate
  Vi s - s s - s


• :::; C log--====
  rmin rminv-logrmin rminv-logrmin
  in the intermediate layer 2r min v - log r min :::; s - s :'.::: r~j~, where

rmin (t) = [1 + o (1)] J2 (n - 1) (T - t).

LEMMA 2.37. Let g (t) be a solution to the Ricci fiow of the form (2.43)

such that 1-iPs I :::; 1 and R > 0 initially. Then there exists C = C ( n, go) such

that

c

IRml :'.::: T-t·

PROOF. By Lemma 2.34, the curvature remains bounded on the polar

caps. On the waist W (t), we obtain the stated bound by combining Corol-

lary 2.24 with Lemma 2.32 D

LEMMA 2.38. The quantity F = 1{, log L evolves by

(

logL-1) (2 - logL) KL;

(2.57a) Ft = b.F + 2 L log L LsFs + log L IJ3

(2.57b) (Vis)

2

• 2P -:;j; -K + L -L + 2QK,