1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

274 24. HEAT KERNEL FOR EVOLVING METRICS

Now define Fk ~ F Fk-l fork EN (F^1 ~ F); since convolution is

associative, we may write F*k = F * · · · * F (k-fold product).

With the parametrix PN defined by (24.24) and satisfying Proposition

24.11, we now follow §2 of the last chapter to establish the convergence of

the associated parametrix convolution series:

00

(24.30) H ~ PN + PN * L (Lx,TPN)*k.

k=l

Using the facts that

00 00

(Lx,TPN) * L (Lx,TPN)*k = L (Lx,TPN)*k

k=l k=2

and that for any GE c^0 (M x M x [O,oo))

(24.31) Lx,T (PN * G) = (Lx, 7 PN) * G - G,

we find that Lx, 7 H = 0. By Lemma 26.4, we also have L;,vH = 0.

Toward verifying (24.31), we first generalize to the time-dependent met-

ric case the properties of differentiating a convolution with the parametrix

discussed in §3 of the previous chapter.

Similarly to Lemma 23.26, we have

LEMMA 24.16 (First space derivatives of a convolution with PN)· Let

(U, {xi} ~= 1 ) be a local coordinate system on M. If

G E C^0 (M x M x JR~) ,

then PN * G is C^1 with respect to the space variables and for x E U, the first

space derivatives of PN * G are given by

(24.32)

8 1

7

axi (PN * G) (x, T; y, v) = v } M { 8PN axi (x, T; z, a-) G (z, CT; y, v) dμg(u) (z) dCT.

SKETCH OF PROOF. Define

(24.33) JN (x, T; y, v; CT)~ JM PN (x, T; z, CT) G (z, CT; y, v) dμg(u) (z),

so that

and

I

8PN 8xi ( x, T,. z, CT ) I -< C ( T - CT )-a rT,X ( z )2a-n-l