1547671870-The_Ricci_Flow__Chow

4. NECKLIKE POINTS IN FINITE-TIME SINGULARITIES 265

Collecting terms as in Lemma 6.34, we obtain

!!..._ F = !:::..F^2 ( l - E) ('VF \l R)

at + R+ p '

_ (T- t)c: ( 2[(R+p)('VR~)-\JRQ9R~\

2

)

(R + p)4-c: 0

+ c (1 - c) 1Rml^2 l'V Rl^2

0 2 2 0

+ 2 (T- t)c: clRml !Rel - P clRml

(^2 4) (T - t)c: pL:
-----..,,.----,,..--+ -------,~
(R + p)3-c: (R + p)^2 - c: (T - t)^1 -c: 3 (R + p)^3 -c:
_::; !:::..F +
2
~;:) ('VF, \l R)

• (T- tr [(2c IRc1^2 - E (R + p)) IR~1^2 + ~pL: - 2P].
  ( R + p) -c: T - t 3
  The result follows when we estimate 1Rcl^2 ::; 1Rml^2 ::; C (R + p)^2 and
  9 ° 27 °
  L: ::; 2 ( i>-1 + JμI + lv l) 1 Rml^2 ::; 2 IRml IRmj^2.
  0
  Notice that the bad terms B1 + B2 = Gip+ C2c IRml come from the
  1/ (R + p)^2 - c: factor in F, whereas the good term - G 1 = -E/ (T - t) comes
  from the (T - t)c: factor. The good term -G2 = -2P would be the only

term present if p = E = 0.

The next two lemmas constitute the second step in the proof of Theorem

9.9 and show that the pinching of the curvature operator improves if there

are no essential necklike points.

LEMMA 9.12. If !Rm -R (e Q9 B)l^2 2 6 IRml^2 for some 6 E (0, 1), then

6 2 0 2

p ;::_ 96 (3 - 6) IRml IRml.

PROOF. We may assume without loss of generality that i>- 1 2 JμI 2 jvj.

The hypothesis implies that

μ2 + v2 + μv ;::_^6

2

(>.2 + μ2 + v2),

and hence that μ^2 + v^2 ;::_ 3 ~ 8 >.^2. Since JμI 2 !v i by assumption, we have

p = >.2 (μ _ v)2 + μ2 (>. _ v)2 + v2 (>. _ μ)2

6

;::_ >.2 (μ - v)2 + 2 (3 - 6) >.2 (>. - v)2 + v2 (>. - μ)2.

Now notice that