1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

9. WEAK SOLUTION FORMULATION 369

Since Dl (0) = 0, given any El > 0 there exists o > 0 such that ll (y)I :::;

Ei lvl whenever IYI :::; o. Hence, if E:::; o, then

a1~ (0) :::; ~ r 11 (z)l ·I a'T/· I (~) dz

8yi En } B(c:) 8zi E E

s Ei r I~ 1 · I a'T/· I (~) d (~)

}B(c:) E 8zi E E

= C1E1,

where c1 ~ JB(l) IYI · I%; I (y) dy is independent of E. This implies

. i· im ale: - (o)
8
. =O.
c:---+O+ yi

(ii) Write

J (x + y) ~ J (x) +DJ (x) · y +~YT· D^2 J (x) · y + l (y).

Then l (y) has first derivative D l ( 0) = 0 and second derivative D^2 l ( 0) = 0.
It suffices to prove that for all (i,j) the mollified function le: (y) satisfies

limc:---+O+ a~:t~j (0) = 0. Note that

Jc: A (y) = n^1 1 J A (z) T/ (z - -y) dz

E B(y,c:) E

and

8

2

le: 0 - ~ r A ( 8

2

'T] ) (~) dz

8yi8yj ( ) - En JB(c:) J (z) 8zi8zj E E^2 •

Since Dl (0) = O, given any Ei > 0 there is a o > 0 such that ll (y)I:::; El lvl^2

when IYI:::; o. Hence when E:::; o,

where c2 ~ JB(l) lvl^2 I ai~~j I (y) dy is independent of E. This implies that

ByiByj^82 fe (0) -- 0 as E ___,. O+.