1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL 227

is bounded independent of (p, y) in any given compact subset](, of Xx M.

Hence, using (23.42) again, we obtain

lim ( PN(x,t;y,u)f(p,x)dμ(x)

t',,u}M

1

n r^2 (x)

= lim 'T/ (x, y) (47r (t - u))-2 e -4(t-u) </>o (x, y) f(p, x) dμ(x)

t',,u B(y,inj(g)/2)

= f (p, y)

since limx-+y </>o (x,y) = 1, where the convergence is uniform in (p,y) EK.

(2)(b) We may similarly prove (23.34b). D

EXERCISE 23.13. Prove that

(1) (~y + JJ P extends continuously to M x M x :IRS,,

(2) PN satisfies (23.34b).

Again, by considering the three cases (i)-(iii) as in part (1) of the proof
of Proposition 23.12, from (23.29), we have

LEMMA 23.14 (Derivatives of heat operator of PN)· For any k,£ EN U

{O},

(23.43)

N n k 2£ d2(x,y)

8f\7~(DxPN)(x,t;y,u) = (t-u) -2-- e-5(t-uJGk,e(x,y,t-u),

where \7~ ~ \7 x o · · · o \7 x ( k times) and Gk,£ is a C^00 covariant k-tensor
on M x M x [O,oo) (note that Gk,£ has support in Minj(g) x [O,oo)).^6 In
particular ( k = £ = 0),

(23.44) ( N-!!:. ( d

2

IDxPNJ(x,t;y,u)=O (t-u)^2 exp - (x,y)))

5 (t-u) ·

PROOF. By (23.33) we have the representation (23.43) for (x, y, t, u) E

Minj(gJ x JR$; recall that 'T/ (s) = 1 ifs~ injJg). Recall also that PN = 0 in

4

(M x M - Minj(g)/ 2 ) x JR~. Thus we only need to consider the case where

(x, y) E Minj(g) - Minj(g)/8·

In this region, one uses (23.38) and the fact that

exp (-(inj (g) /^8 )

2

) · (t - u)-p--+ 0

20 (t - u)

as t - u--+ 0 for any p E JR. Since the </>k are bounded in Minj(g) - Minj(g)/8
for k = 0, 1, ... , N, by definition (23.9) we have that for (x, y, t, u) E

(^6) Since we only need a little bit of an exponential to dominate a polynomial, in the

RHS of (23.43) we may replace the factor 5 in the denominator of the exponential by 4 + c

for any c > 0.