1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

298 24. HEAT KERNEL FOR EVOLVING METRICS

where C < oo is independent of k, f3k, /k and where we used d-p'Yk s CJ'k/^2 d-;'Yk

/2.

and dμg(p) S c; dμg(T) by (24.95).

We break up the region of integration 8M into three subregions:

R1 ~ {w E 8M: dT (w,xo) s !dT (xo,z)} = Bg(T) (xo, !dT (xo,z)),

R2 ~ { w E 8M : dT (w, z) s !dT (xo, z)} = Bg( 7 ) (z, !dT (xo, z)),

R,3 ~ { w E 8M : d 7 (w, xo) 2: !d 7 (xo, z) and dT (w, z) 2: !dT (xo, z)},

where the balls are contained in 8M.

Since d 7 (w, z) 2: !dT (xo, z) for w E R1, we have

r d:;n+^2 a (xo, W) d:;"fk (w, z) dμg(T) (w)

ln1

:S 2'Ykd:;'Yk (xo, z) 1 1 d:;n+^2 a (xo, w) dμg(T) (w)

Bg(r) ( xo, 2dr(xo,z))

(24.99) s C2'Ykd;a-'Yk-l (x 0 , z),

where C < oo is independent of k, /3k, "/k (the region of integration is an

(n - 1)-dimensional ball).

Since dT (xo, w) 2: !d 7 (xo, z) for w E R2, we have

(24.100)

r d:;n+^2 a (xo,w)d:;"fk (w,z)dμg(T) (w)

ln2

:::;2n-2ad:;n+2a(Xo,z) r ( 1 )d:;'Yk(w,z)dμg(T)(W)

}Bg(r) z,2dr(xo,z)

• < + C 1 d2a-'Yk-1 T (x Q, z) >
  -/k n-
  where C < oo is independent of k, f3k, /k and where we used 'Yk s n - 2a <

n-1.

Since d 7 (xo, w) 2: ~dT (w, z) for w E R,3 and R,3 c 8M - R2, we have

r d:;n+^2 a (xo, W) d:;'Yk (w, z) dμg(T) (w)

lns

S 3n-2a r d:;n+2Cl'.-"fk (w, z) dμg(T) (w)

hM-R2

~

diam(M,g(T))

< C r^2 a-'Yk-^2 dr

• (^1) 2dr(xo,z)
  _ - C r 2a-'Yk-11diarn(M,g(T)) 1
  2a - "fk - 1 2dr(xo,z)
  2'Yk-2a+ic

< d2a-'Yk -1 (x z)

(24.101) - fk - 2a + 1 T Q,

provided /k > 2a - 1, where we used the fact that 8M is compact and the
volume comparison theorem in 8M - R 2.