1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

220 23. HEAT KERNEL FOR STATIC METRICS

Nate that for r < inj (g) we have that a is bounded and that

(23.19) 8loga 8r -- 8r^8 1 og v ~ ue~g-- -r-n -1 -- 0 (1)

for r near 0.^4

Applying the heat operator to (23.9) and using the defining equation

(23.11) and also (23.18) yields

(23.20)

N N

_ r 8log a "°""' k '°' k

--2(t-u) 8r EL</>k(t-u) +2L._.,,(t-u) (\JxE,\J</>k)

k=O k=O

N N

+EL 6-x</>k · (t - u)k - EL k</>k (t - u)k-l.

k=O k=O

Note that

(23.21) ( r )E-8^8 E TxM.

2 t-u r

We may thus rewrite (23.20) as, after factoring out E and cancelling the

term on the LHS,

N N

0 = _'!:._ 8log a'°' </>k (t - u)k-l - '°' (t - u)k-l r 8</>k

2 ~ L._.,, L._.,, ~

k=O k=O

N-l N

• L .6.x</>k · (t - u)k - L k</>k (t - u)k-l;
  k=O k=O
  that is, by grouping like terms, we have

(23.22) 0 = - (t - u)-^1 r (! 8loga </>o +^8 </>o)

2 8r 8r

N

+ L._.,, '°' ( -r 8</>k 8r - 2 r 8log a ) k^1

k=l^8 r </>k - k</>k + 6-x</>k-l (t - u) -.

Thus we have established the first part of

LEMMA 23.7 (Recursive ODEs for <i>k)· Let (Mn,g) be a closed Riemann-

ian man if old and let y E M.

(^4) By definition, <p (x) = o (l) iflimx-J-y ~f;j = 0.