1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. LOCALIZED NO LOCAL COLLAPSING THEOREM

Hence, for a minimal £-geodesic"( from (x,O) to (w,r), we have

L(w,r)=2v'T for vs(Rg('Y(s),s)+b'(s)i;(s))ds

2: 2y'T for Vs ( 2 (ln-s)) ds

• -2n if T < -2 !.

(3) This part follows directly from (2), which implies for w E M

• 1
  L(w,

2

)+2n+l 2: 1.

75

(4) Let u = dg(t)(xo, w)-(2t-l)A, d = dg(t)(xo, w), and L = L (w, 1-t).

We compute

(18.70a) ah at (w, t) = </>'(u) (ad at - 2A ) (L -+ 2n + 1) + </>(u)at' aL

(18.70b) \Jh(w, t) = (L + 2n + 1)</>'(u)\Jd + <f>(u)\JL,

Ah(w, t) = (L + 2n + 1) <f>'(u)Ad + (L + 2n + 1) <f>"(u) JVdJ^2

+ 2 (V</>(u), VL) + </>(u)AL

(18.70c) = (L + 2n + 1) </>'(u)Ad + (L + 2n + 1) <f>"(u)

+ 2 (V</>(u), VL) + </>(u)AL.

Hence

( ~~ - Ah) (w, t) = </>(u) (:t - A) L - 2 (V</>(u), VL)

(18.71) + (L + 2n + 1) (-</>"(u) + </>'(u) ( ~~ - Ad-2A)).

Given t, at a minimum point y of h( ·, t) we have \Jh = 0. Hence, by

(18.70b), we have at (y, t)

</>(u)\JL = -(L + 2n + 1)</>'(u)\Jd

and hence

• 
  
  
  
  • (</>'(u))2
    (V</>(u),VL) = -(L+2n+l) </>(u).
  
  
  
  

It now follows from (18.71) that

(~~-Ah) (y,t) = </>(u) (gt -A) L

(18.72) + (L + 2n + 1) ( -</>"(u) + ifuJ (</>'(u))

2

).

+ (~~ - Ad-2A) </>'(u)

Note that fort 2:! we have dg(t)(w, xo) 2: !el-n whenever </>'(u) f 0

at (w, t). From Bg(t)(xo, e^1 -n) C Bg(o)(xo, 1) for all t E [O, 1] and from