1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. HEAT KERNEL FOR AN EVOLVING METRIC

we have

Id~ (JM H (x, r; y, v) dμ 9 ( 7 ) (x)) I ::S C1 JM H (x, r; y, v) dμ 9 ( 7 ) (x).

On the other hand, by taking¢= 1 in (26.8), we have

lim r H (x, r; y, v) dμg(T) (x) = 1.

T\,v}M

Hence we have

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LEMMA 26.5 (L^1 -norm of heat kernel on a closed manifold is bounded).

If M is closed, then the heat kernel for Lx, 7 satisfies

(26.19) e-G1(T-v) :::; JM H (x, r; y, v) dμg(T) (x) :::; eG1(T-v)

for any y E M and 0 :::; v < T :::; T.

If Q = R, then C1 = 0, so that (26.19) yields the following.

COROLLARY 26.6 (L^1 -norm of heat kernel on a closed manifold is pre-
served when Q = R). If M is closed and Q = R, then

(26.20) JM H (x, r; y, v) dμ 9 ( 7 ) (x) = 1

for any y E M and 0 :::; v < T :::; T.

On the other hand, by Lemma 26.4 we have

Id~ (JM H (x, r; y, v) dμg(v) (y)) \

so that

=\JM(~~ (x,r;y,v) +R(y,v)H(x,r;y,v)) dμg(v) (y)\

= \JM (-!:ly,vH (x, r; y, v) + Q (y, v) H (x, r; y, v)) dμ 9 (v) (y) \

::S sup IQI \JM H (x, r; y, v) dμ 9 (v) (y)\,

LEMMA 26.7 (L^1 -norm of heat kernel using the second space vari&bles).

If M is closed, then the heat kernel for L satisfies ·

(26.21)

where C2 ~ SUPMx[O,Tj IQ!.