1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL 225

a cutoff function to obtain a parametrix for the heat operator. Define the

function

by

(23.35) PN (x, t; y, u) ~ rJ (d (x, y)) HN (x, t; y, u),

where rJ : [O, oo) --+ [O, 1] is a C^00 radial cutoff function with

(23.36) rJ (s) = { -^4 '

1 if S < inj(g)

0 if S 2: injJg).

Note that rJ (x, y) ~ rJ (d (x, y)) is C^00 on all of M x M. Moreover, we may

assume that rJ is such that there exists a constant C < oo depending only

on inj (g) such that

l\7 xrJI :S Cy'ij :S C and l.6.xrJI :S C

on M x M. We also have bounds for all of the.higher derivatives of r], i.e.,

there exist Ck< oo such that on M x M

1\7~r]1 :::; ck fork 2: 2.

Since HN is C^00 on Minj(g) x :IRS, and the support of rJ (d (x, y)) is con-

tained in Minj(g)' we conclude that PN E C^00 (M x M x :IRS,). Note that

(23.37) JM IPN (x, t; y, u)I dμ (y) :::; C

for some constant C < oo.

PROPOSiTION· 23.12 (Existence of a parametrix for the heat operator).

If N > n/2, then PN is a parametrix for the heat operator .6. - gt.

PROOF. (0) We have already shown that PN E C^00 (M x M x :IRS,).

(1) We have

(23.38) DxPN = r]DxHN + (.6.xrJ) HN + 2 (\7 xrJ, \7 HN).

To show that DxPN extends continuously to M x M x :IRS,, we estimate the
RHS of (23.38) on M x M x JRi, where JR~ is defined in (23.30). Observe that
we may estimate DxPN in three relatively open sets covering this region as
follows.

(i) In the exterior region (M x M - Minj(g); 2 ) x :IRS, we have

PN := 0 and DxPN := 0.

(ii) In. the interior region Minj(g)/4 x :IRS, we have

DxPN = DxHN.

Recall that DxHN satisfies (23.28) and extends continuously to a function

defined o~ Minj(g) x :IRS,, which takes the value O on Minj(g) x 8 (:IRS,).