1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

138 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS

we obtain

(30.17)

• 1 w=-~ d2 (q,p=)

L i (q, t) ~ 4K (t) VWi - t goo(to) _ 2 dt

f (w=-t)

• r i VWi - t sup Rgi (x, t) dt.

lt xEMi

By (30.12) and r < ~'we h ave

1

wi n2M - 3

VWi - t sup R 9 i (x, t) dt ~ - 3 - (wi - t )^2 -r

t xEMi 2 - r

n2M - ~-r

~ -3 - (w= -t)2.

2-r

Applying this to (30.17) while using w= - ~ - t = w"2-f and K (t) ~ K(w= - ~)

for t E [f, w= - H we find that for all q E M= and t E [to, w=)

(30.18)

• 
  
  
  
  • ( €) d~ 00 (to) (q,p=) n^2 M - ~-r
  
  
  
  

Li(q, t) ~ 2K w= - -

2

vw= - a = + - 3 - (w= - t).

W= - t 2 - r

This implies (30.9) if r > 1 and (30.10) if r = 1. The proof of Lemma 30.4 is
complete. D

EXERCISE 30.6. Determine if one can qualitatively strengthen the estimate

(30.9) by considering test paths"( with 'Yl[w= - o,wi) =Pi , where 8 E (0 , c:) is chosen

optimally. Note t hat in the proof of Lemma 30.4 we chose 8 = ~-

1.4. First derivative bounds for the reduced distances.

Next we establish bounds for the gradients and time derivatives of the Li, which

are uniform (i.e., indep endent of i) on compact subsets of M= x (a=,w=)·

LEMMA 30.7 (Uniform bounds for the gradient of Li on compact sets). Assume

the same hypothesis as in Lemma 30.4. Then, for any compact set K C M= and

any c: E (0 , w=;°'= ), there exists const < oo such that f or any q E K and any

t E [a=+ c:, w= - c:], we have

for all i sufficiently large.

Idea of the proof. The gradient of the £-distance is basically given by the

tangent vector of a minimal £-geodesic. Hence we can estimate its norm by the

£-geodesic equation with the aid of the Type A condition, Shi's estimate for IV RI,

and the previous pointwise bound for the £-distance.

PROOF. STEP l. The standard formula for IV' Lil:g.- Let q E M= and let