1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. HEAT KERNEL FOR AN EVOLVING METRIC 343

PROOF OF LEMMA 26.14. (1) Upper bound. Let {DihEN be an exhaus-

tion of M as above and let Hni (x, r; y, v) be the Dirichlet heat kernel of

(ni,g (r)) as in (24.107), i.e.,

_lim Hni (x,r;y,v) = H(x,r;y,v).

i--+oo

By the upper bound (26.27), with 'M =Di', we have

r Hni (x,r;y,v)dμg(T) (x) s eCi(T-v)

lni

for any y E int (Di) and 0 S v < T S T. Since Hni converges to H, we

conclude

(26.39) JM H(x,r;y,v)dμ 9 ( 7 ) (x) S eCi(T-v)

for any y E M and 0 S v < T S T.

(2) Lower bound. We prove a lower bound for the integral on the

LHS of (26.39). Let ¢ be the cutoff function defined in (25.85). Since

supM !sect (g (0))1, SUPMx[O,T] IRijl, and SUPMx[O,T] IY'iRjkl are finite, by

(25.86), (25.94), and (25.95), we have

(26.40) 1~g(T)¢I s Vn JV'Y'¢lg(T) s Vn (CV</) ~ + R2 C) s R Cn

for some constant Cn < oo independent of R, where we assume R 2 1. We

compute for any r E (0, T)

Id~ JM ¢(x)H(x,r;y,v)dμ 9 (7 ) (x)I

= IJM ¢ (x) ( ~~ (x, r; y, v) + R (x, r) H (x, r; y, v)) dμg(T) (x)I

S IJM ¢(x) (~x,TH) (x,r;y,v)dμ 9 ( 7 ) (x)I

+ IJM ¢ (x) (R-Q) (x, r) H (x, r; y, v) dμ 9 ( 7 ) (x)I

S IJM H (x, r; y, v) (~x, 7 ¢) (x) dμ 9 ( 7 ) (x)I

• C1 JM¢ (x) H (x, r; y, v) dμg(T) (x).

Applying (26.40) and (26.39) to this, we obtain

__!},_ ( cp(x)H(x,r;y,v)dμg(T)(x)

dr }M

;::: -; eCi(T-v) - C1 JM¢ (x) H (x, r; y, v) dμ 9 ( 7 ) (x).