- DIFFERENTIAL HARNACK ESTIMATES OF LYH TYPE 147
PROOF. As in Lemma 5.55, we computeat fj Q A = ( R - r) b.L A + b. ( b.L A + I \7 L A^12 + R - r + s )
+ ( R - s) ( b.L + I \7L 12+ R - r + s) + s ( s - r)
= b.Q + 2 ( VQ, \7 L) + 2 lvv £ 1
2
+ 2 (R - r) b.L + (R - r)^2- s I \7 LI 2 + ( r - s) ( b.L + R - r) + s ( R - r).
D
Our aim is to obtain a lower bound for Q. This is accomplished in the
following estimate.
PROPOSITION 5.60. Let (M^2 , 9 (t)) be a solution of the normalized Ricci
flow on a closed surface. Let 9o be any initial metric such that r > 0. Then
there exists a constant C > 0 depending only on 90 such that
Q 2: -C.
PROOF. By (5.28) and Proposition 5.18, there is C > 0 depending only
on 9o such that -ce-rt :S s :S 0 and R :S Cert. Hence the only possi-
bly troublesome negative term on the right-hand side of (5.42) is s l\7 L l^2 ,
over which we have no control. However, we can avoid this bad term by
considering the quantity
p ~ Q + sL.
Indeed, it follows from (5.4) and (5.41) that
:t ( sL) = b. ( sL) + s I \7L 1
2
+ s ( R - r + s) + ( s - r) ( sL).Rewriting the gradient term and using the fact that L = log (R - s) >
- c (1 + t) for some c > 0, we see that there is C1 > 0 such that
:t (st) 2: b. (st)+ 2(\7 (st), vi) - s lvLl
2- C 1.
By adding the evolution equations for Q and sL and recalling the standard
fact that n IS l^2 2: (tr 9 S)^2 for any symmetric 2-tensor S, we obtain:t P 2: b.P + 2 ( \7 P, \7 i,) + Q^2 + ( r - s) Q - C 2
for some C2 > 0. Since sL = P - Q is bounded, it follows that there exists
a constant C > 0 such that:t P 2: b.P + 2 ( \7 P, \7 i,) + ~ ( P^2 - C^2 ).
Applying the maximum principle, we getQ + sL = P 2: min {min P (x, 0), -c},
xEM^2