1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

326 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

wherever P < 0, we obtain from (25.75) that

(25. 78)

0;::: 2-3c </>P2

n

+ (8</> - ~</> + (2+15-1) l\7¢12 + 1!_ - (2 (2 - 3c) c - 6) </> l\7 Ll2) p

~ </> ~ n

+ </> (

2

~

3

c c^2 l\7Ll

4

• 01 l\7Ll

2

• 02).

Now we take 6 =^2 C^2 ~^3 e) c, which is positive since c E (0, 2/3).^4 Suppose
that the first two terms on the second line of (25. 78) satisfy

(25. 79) 8¢ ( n ) 1'7</>1

2

OT-~</>+^2 + 2(2-3c)c -</>-S C^3 ,

where 03 E [O, oo) is to be determined after we choose </> in (25.85) below.
Then we have at (xo, To), where T</>P attains a negative minimum,

(25.80) O ;::: 2 - 3c ( <f>P)2 + (03 + ±) </>P - </>2 (02 + nCi ) '

n To (2 - 3c) c^2

where we multiplied the whole equation by </> and where we also used

a I \7 L 14 - 01 I \7 L^12 ;::: --Ci a

for a> 0.

We have a quadratic inequality in </>P, which we use to bound P. Note
that if there is a number x E IR satisfying an inequality of the form

ax^2 +bx + c S 0,

where a > 0, b ;::: 0, and c < 0, then b^2 - 4ac > 0 and we have the lower

bound
-b-Jb^2 -4ac b+ ~
x> - 2a >-- a.

Hence

To¢P? - 2 :" 3£ ( ToC3 + ¢ + To</>

2

~" ( c, + (2"~'')).

Furthermore, since To S f' S T and 0 S </> S 1, we have
(25.81)

To (¢P) (xo, To)? - 2 :" 30 ( 1+TC3+7

2

~" ( C2 + ( 2 "~'')) °"' -6.

(^4) More generally, we may take J E (0, (^2) C (^2) : (^3) "'l c] since then on the second line of (25.78)
the term - (^2 <^2 :^3 "') E: - J) ¢>IV Ll^2 P is nonnegative.