1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

230 23. HEAT KERNEL FOR STATIC METRICS

For basic properties and results about general convolution transforms,

see Hirschman and Widder [100].

2.2. The parametrix convolution series.

Since PN depends only on x, y, t - u, accordingly, we also write

(23.50) PN (x, y, t - u)-:-PN (x, y, t, u).

Recall from Proposition 23.12 that PN E C^00 (M x M x (0, oo)) for N > ~

defined in (23.35) is a parametrix for the heat operator D ~ ~ - gt.

Applying the heat operator to (23.46), while using the formulas (23.93)

and (23.i09) below, we have for GE c^0 (M x M x [O, oo))

Dx (PN * .. G) (x, y, t)

, : · ~ -G (x, y, t) +lat JM (DxPN) (x, z, t - s) G (z, y, s) dμ (z) ds.

Therefore we have the following formula relating convolution and the heat

operator.·

LEMMA 23.15 (Heat operator of a convolution with the parametrix). For

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

(23.51) ' , ' Dx (PN * G) = (DxPN) * G - G,

where ?;. is the parametrix defined by (23.35).

Now 'we construct the heat kernel using convolution. We look for a heat

kernel of the form

(23.52). '{" ' H ~ PN + PN * G,

where GE c^0 (M x M x [O, oo)) is a function to be determined. By (23.51)

we have

(23.53) ·.·

Hence'dxH = 0 is equivalent to

(23.54) '

Making the analogy of this equation with the equation x + xy - y = 0,
whose,·sulution is
00
y = x = '"'"'xk,
1-x ~
k=l

we see that the parametrix convolution series

00

(23.55) GN ~ L (DxPN)*k

k=l

is a fo:mai' solution of (23.54).