324 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS
PROOF. Applying the assumed bounds in (25.69) to the evolution equa-
tion (25.67), we obtain^3
~: ~ l:!:.P + 2 ("vL, VP)+ 2 (1-c:) IVVLl^2 - 2A IVVLI
- 2 ((1-c:) K +Ac:) IVLl^2 - C' IVLI - C'.
Then applying the Peter-Paul inequality ax^2 - bx ~ -1: to both the first
and second derivatives of L, we have
8P A^2
-
8
~ f:!:.P + 2 (VL, VP)+ (2 - 3c:) JVVLJ^2 - -
T c- 2 ((1-c:) K +Ac:+ C') JVLl^2 - 2C'.
Finally, applying IVV LJ^2 ~ ~ (!:!:.L )^2 to this, we obtain the corollary. D
Note that we are trying to obtain a lower bound for P, so that the term(^2) - (^310) (l:!:.L)^2 on the RHS of (25.70), which is similar to P (^2) , is a 'good' term.
n.
2.5. Localizing the Harnack calculation.
To apply the maximum principle to the evolution inequality for P when
Mis noncompact, we need to 'localize' the above calculation of the evolution
of P, i.e., multiply P by a cutoff function so that it has compact support.
First we make some calculations using a general cutoff function and then we
choose a suitable cutoff function.
'Let RE [1, oo) and let</>: M x [O, T]----+ [O, 1] be a C^2 function with
(25.72) supp(</>) c Bg(O) (p,2R) x [O,T].
Multiplying the evolution inequality (25.70) for P by T</> (the factor T is
motivated by Remark 25.16), since</> is independent of time, we have at any
(^3) Note that since c E (0, 2/3),