1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

382 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

Once we have (26.145), we may apply the maximum principle of Karp

and Li (see the proof of Theorem 1 in [106] or Theorem 1.2 in [144])^12 to

obtain (26.142).

Now we turn to the proof of (26.145). Given R E [1, oo ), let 'f} : M -+

[O, 1] be a C^00 cutoff function with^13

'fJ ( x) = 1 if d ( x, 0) :::; R,

'f} (x) = 0 if d (x, 0) ;::::: 2R,

IY''IJl^2 /'fJ (x) :SC/ R^2 if d (x, 0) < 2R,

where C < oo is a universal constant. Using (26.143) and integrating by

parts, we have

fo

1

JM e-ar

2

(x)'I] (x) l\7f1^2 (x, t) dμ (x) dt

= -~ fo

1

JM e-ar

2

(x)'I] (x) ( :t - ~) (!^2 ) (x, t) dμ (x) dt

::=:; C - f

1

f f (x, t) (\7 (e-ar

2

(x)'I] (x)), \7 f (x, t)) dμ (x) dt,

lo 1B(0,2R)

where we used

Since

~JM e-ar2(x)'I] (x) (!2) (x, 0) dμ (x)

• ~JM e-ar2(x)'f} (x) (!2) (x, 1) dμ (x)

:S ~JM e-ar

2

(x)'f} (x) (r (x) + 1)^2 dμ (x)

:::; c.

- f 1 f f (x, t) I \7 (e-ar

2

(x)'f} (x )) , \7 f (x, t)) dμ (x) dt

lo j B(0,2R) \

::=:; -

2

1

f 1 f e-ar

2

(x)'I] (x) J\7 f J^2 (x, t) dμ (x) dt

lo 1B(0,2R)

1111 ear2(x) 2

+ - 2 -( -) 1\7 (e-ar

2

(x)'I] (x)) I f^2 (x, t) dμ (x) dt,

0 B(0,2R) '/] X

(^12) See also Theorem 12.22 on pp. 153-154 in Part II.
(^13) We only need that fJ is Lipschitz.