- LOCAL ESTIMATE FOR THE SCALAR CURVATURE UNDER RICCI FLOW 37
so that(28.10)
(
f) (8R ) r/ o i. (or )
= 'T/ o p 7ft - 6.R + R-;- at - 6.f- -2( 'T/^1 o '!':_ -) ('Vf, 'V R) - __ 'TJ"oi. PR
p p p2
since !Vfl
2= l.
STEP 4. An ordinary differential inequality (oDI) for Smin· Let Smin (t)
minyEM S (y, t). Note that(28.11) Smin (to):=:; R(O,to) < 0.
For any t E [a, to] where Smin (t) < 0, let Xt be such that S (xt, t) = Smin (t). Note
that R (xt, t) :::; 0. We consider two cases while calculating at the point (xt, t).(1) If dg(t) (xt, 0) :::; ro, then f (xt, t) :::; p. This implies ( 'T/ o ~) (xt, t) = 1 and
( 'T/^1 o ~) (xt, t) = ( 'T/^11 o ~) (xt, t) = 0. Thus (28.10) says that at (xt, t) we have
(28.12) ( -^8 - 6. ) S = 8R - - 6.R > -^2 R^2.
at at -n
(2) If dg(t) (xt, 0) 2 ro, then we may apply (28.7), 'TJ^1 :::; 0, and ~~ 2 6.R+ ~R^2
to obtain that at (xt, t)(28.13) (-^8 - 6. ) S^2 ( 'TJO '!':_ -) -R^2 2 - ___ 2r/of. P ('Vf, 'VS)
& p n p 'T/1 ( ( 'T/1
O ~ )2
+- 2 -TJ II o-f) R.
p2 'T/ pNow take p = 4 in (28.2). Using (28.12) and applying (28.3) to (28.13), we
have at (xt, t) in either Case (1) or Case (2) that
(28.14) ( -^8 - 6. ) S 2 ( 'T/ o -f) -^2 R^2 + --const ( 'T/ o -f) ~ R ,
& p n ~ pwhere const > 0 is a universal constant and where we used 'VS (xt, t) = 0.
By (28.14) and (28.8), for any€ E (0, ~) we have(28.15) d_S mm. () t 2 ~ 'T/ (r(xt,t)) R( Xt, t )^2 + const 2 'T/ ~ (r(xt,t)) R( Xt, t )
dt n p p p(
2 ) (r(xt,t)) R( )^2 const
2
> - - € 'T/ Xt t - --- n p ' 4cp^4
(
- -^2 € ) s. (t)^2 - --· const^2
- n mm 4cp4 '
here we have used ab 2 -rn^2 - 41 e:b^2 and 0 :::; 'T/ :::; 1, where ~t denotes the lim inf
of backward difference quotients.