1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

330 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

since ~~ = 0 and by (25.95) and (25.94). By our definition of C3 in (25.96),
we finally obtain (25.79).

By the lower bound for f in (25.90), the cutoff function ¢ defined by

(25.85) satisfies

supp(¢) CBg(o)(p,2R-1) x [O,T].

On the other hand, by the upper bound for fin (25.90), we have

(25.97) ¢ = 1 in Bg(O) (p, R-Cn,K) x [O, T].

Hence, by (25.82) we have

(25.98)

c

P(x,r) 2 --

r

for (x,r) E Bg(o)(p,R-Cn,K) x (0,7] and for any r E (0,T].
In particular, given any (x,r) E Bg(o)(p,R-Cn,K) x (O,T], by taking
r = r, we obtain from (25.98) and (25.81),

(25.99) P(x,r)2---n (1 -+C3+

2-3c: r

2-3c: -n-(c 2 + (2-3c:)c:2 nCi )) ·

Hence, using (25.71), we obtain

P(x r) > __ n_ (_!+Cs+ C5 +c1)

' - 2 - 3c: r R R^2 '

where the dependences of Cs, C5, and C7 are exactly as in the statement of

Theorem 25.8. Hence we obtain (25.53); this completes the proof of Theorem

25.8.

2.8. The case of the Ricci fl.ow.

We now complete the proof of Theorem 25.9, where Rij = -Rij. Sup-

pose for each r E [O, T] we have Re ( ·, r) :S (n - l)K in Bg(r) (p, 2 R) for

some constants K 2 0 and R 2 JR. Let

f (x, r) = dg(r) (x,p).

Then [\7 f [^2 = 1 and by (18.8) we have

(

Bf \ 5

Br - b.f) (x,r) 2 -3(n-l)VK

for all x E Jvl - Bg(r)(P, )K).

By (25.89) we now have

B¢ -b.¢+ (2+ n ) [\7¢[2

Br 2 (2 - 3c:) c ¢

(25.100) <---(n-l)vK+ y10¢5 rr:; ( 3+ n ) -='=C3 C -

• R 3 2 (2 - 3c:) c: R^2.