1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

272 24. HEAT KERNEL FOR EVOLVING METRICS

where Fk,£ is a C^00 covariant k-tensor on UrE[O,T] Minj(g(r)) x { T} X [O, T].

EXERCISE 24.9. Prove the above lemma.

Now let iJ : [O, oo) --+ [O, 1] be ·a nonincreasing C^00 cutoff function with

iJ (s) = 1fors:::;1 and iJ (s) = 0 for s;::::: 2. Define p ~ ~ min 7 E[O,T] inj (g (T)).

Given HN as above, define the parametrix

PN : M x M x JRf --+ JR

by

(24.24) P N ( x,y,·T,V ) 'T'T/. -(d^7 (x,y))H p N ( x,y,T,V. )

Analogous to Definition 23.11 we have

DEFINITION 24.10 (Parametrix for Lx, 7 ). We say that a C^00 function

P : M x M x JRf --+ JR is a parametrix for Lx,r if

(1) the functions Lx,rP and L;,vp both extend continuously to M x

M x JRf, where

L;,v ~ :'U + f::::.y,v - Q + R,

and

(2) lim 7 \,v P ( ·, T; y, v) = 8y and limv/'r P (x, T; ·, v) = 8x, that is, for

any function f E c^0 ( M),

(24.25a) lim r p (x, T; y, v) f (x) dμ (x) = f (y),

r\,v}M

(24.25b) lim r P(x,T;y,v)f(y)dμ(y)=f(x).

v/'r}M

By essentially the same proof as for Proposition 23.12, one can show the

following.

PROPOSITION 24.11 (Existence of a parametrix for Lx, 7 ). If N > n/2,

then PN is a parametrix for Lx,r·

Moreover, analogous to the Lemma 23.14, we have

LEMMA 24.12 (Derivatives of Lx,rPN)·

(24.26)

£ k N n k 2£ -i;_(x,y)

8t'Vx(Lx,rPN)(x,y,T,v) = (T-v) -2-- e^5 (-r-v) Gk,e(x,y,T,v),

where Gk,£ is a C^00 covariant k-tensor on M x M x JRf. In particular

(compare with (23.44)),

(24.27)

for some constant Co < oo.^2

(^2) Similarly to as in the previous chapter, one may replace the factor 5 on the RHS of

(24.27) by 4 + s for any s > 0.