1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

336 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

in (26.11). Since (LxH) (x,7;z,p) = 0 and (L;H*) (x,7;y,v) = 0, we have

for p < 71 < 72 < v

0=1T

2

d7 { (Lx,TH) (x,7;z,p)H* (x,7;y,v)dμg(T) (x)

Tl JM

-1T

2

d7 r H (x, 7; z, p) (L;,TH*) (x, 7; y, v) dμg(T) (x)

Tl JM

(26.13) =JM H(x,72;z,p)H* (x,72;y,v)dμg(T2) (x)

• JM H (x, 71; z, p) H* (x, 71; y, v) dμg(Tl) (x).

We then take the limits as 71 .c p and 72 /" v in (26.13), while using
(26.7) and (26.10), to obtain
LEMMA 26.3 (Symmetry between heat and adjoint heat kernels on a
closed manifold). For any y, z EM and 0 :Sp< v :ST we have

(26.14) H(y,v;z,p) =H*(z,p;y,v).

We can now prove the following.
LEMMA 26.4 (For second set of space-time variables His the heat kernel
for L*).

(26.15)

(26.16)

( ~~ + .6.y,vH) (x, T; y, v) + (R - Q) (y, v) H (x, 7; y, v) = 0,

lim H (x, 7; ·, v) =Ox.

V ,/'"T.

That is, with respect to the second pair of variables y and v, H is the fun-

damental solution to L*u = 0.

PROOF. Substituting (26.14) into (26.9)-(26.10), we obtain (26.15)-

(26.16). D

Note that using (26.6) and (26.3), we have for any y E M and 0 :S v <

7 :ST that^2

! (JM H(x,r;y,v)dμg(T) (x))

=JM(~~ (x,7;y,v) +R(x,7)H(x,7;y,v)) dμg(T) (x)

(26.17) = JM((R(x,7)-Q(x,7))H(x,7;y,v))dμ 9 (T)(x).

Assuming that

(26.18) sup \R-Q\ ~ C1 < oo,

Mx[O,T]

(^2) An elementary justification of the interchange of the time derivative and the space
integral is given by Lemma 23.40.