1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. BOUNDS OF THE HEAT KERNEL FOR AN EVOLVING METRIC 357

PROOF. We only prove the inequality

H(x,T;y,v)::; ( ~)

Vol_g B_g x, y ~- 2 -

and leave the other one as an exercise. Let <T ~ fl and r ~ d_g ( x, y).

(1) If <T .2:: f, then we may invoke Lemma 26.17 to obtain

c

H(x,T;y,v):SVil-B-( 0 g g x, (T )

1

Ce2as. - r2

< e 2050-2

• Vol_g B_g (x, <T)

for any constant C5 > 0.

(2) Suppose <T < f. Recall that (26.65) says

C3 exp ( - 2 6:cr2 )

H(x,T;y,v)::; 1/2 1/2.

Vol 9 B_g (x, <T) · Vol 9 B_g (y, <T)

(26.66)

Since B_g (x, <T) C B_g (y, f + <T), regarding the RHS of (26.66), we have

-1 _ 1 ( Vol_gB_g(y,r+<T)

Vol 9 B_g(y,<T):SVol 9 B_g x,<T) Vil-B-( 0 g g y, (T )

Vol_...K__ B (r + <T)

:S Vol_§l B_g (x, <T) Vol~~ B (<T) '

n-1

where we used the Bishop-Gromov relative volume comparison theorem and
where Vol_...K_ B (r) denotes the volume of a ball of radius r in the simply-
n-1
connected space form with constant sectional curvature - n1:. 1. Thus

1

C 3 exp (-2C4cr r^2 2 ) Vol--^2 K B (r + <T)

(26.67) H (x, T; y, v) :S ( ) n;l

Vol_g B_g x, <T Vol2 K B (<T)

-n-1

For all <T,r > 0 such that <T::; max{r, A}, we have

(

r2 ) Vol_...K__ B (r + <T)

exp - 2C4<T^2 Vol~~ B (<T)

n-1

(

r2 ) Vol_...K__ B (2r)

<ex --- n-1

• P 2C4<T^2 Vol_...K__ B (<T)
  n-1
  (26.68)