1547671870-The_Ricci_Flow__Chow

140 5. THE RICCI FLOW ON SURFACES

When the initial scalar curvature is nonnegative, these two lemmas lead
to the following observation.

COROLLARY 5.49. Let (M^2 ,g(t)) be a solution of the normalized Ricci

flow on a closed surface with R(-,0) 2: 0. Then for any to E [O,oo), the

estimate

(5.32)

1

-g (x, to) ::; g (x, t) ::; veg (x, to)

e

holds for all x E M^2 and t E [to, to+ 2 Rm~x(to)] ·

9.2. Estimating the gradient of the scalar curvature. If the scalar
curvature R is bounded above by some constant "' on a time interval [O, T),
then applying the maximum principle to (5.20) proves that

IY'Rl2::; e4"t-3J;r(f)dt sup IY'Rl2 (x,O).

xEM^2

However, we prefer a bound for IV Rl^2 which depends only on the initial
scalar curvature R (-, 0) and not on its derivatives. Of course, any such
bound must blow up as t "'>, 0.

PROPOSITION 5.50. There exists a universal constant C < oo such that

for any solution (M^2 , g ( t)) of the normalized Ricci flow on a compact sur-

face that satisfies r 2: 0 and IR (-, 0) I ::; "' for some "' > 0, the estimate

sup IV RI (x, t) ::; C"' r,.

xEM^2 vt

holds for all times 0 < t ::; 1/ (CK,).

PROOF. The idea is to compute the evolution equation of

G ~ t IY'Rl^2 + R^2 ,

and apply the maximum principle. Notice that G^2 ::; R^2 ::; "'^2 at t = 0. It
follows from (5.20) that

:t (t IV R1

2

) = ~ (t IV Rl

2

) - 2t IV\7 Rl^2 + [t (4R - 3r) + 1] l\7 R l^2.

Adding this equation to

:t (R^2 ) = ~ (R^2 ) - 2 IV' Rl^2 + 2R^2 (R - r)

shows that

(5.33) :t G::; ~G + (4tR - 1) IV' R l^2 + 2R^3.

In order to apply the maximum principle to (5.33), we need to estimate IRI
on a suitable time interval. Since

a

otR2:R(R- r)