1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

308 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

on n x [O, T]. We shall integrate this inequality after localizing it (to enable

integration by parts).

Let 0 :S r1 < r2 :S T and let

(25.13)

be a cutoff function with support contained in D x [ri, T2], where D c n is
a compact regular ( C^1 ) subdomain. Assume that

(25.14) 'ljJ (x, Tl) = 0

for all x E n. We shall discuss the choices of D and construction of the

corresponding 'ljJ and bounds for their first derivatives later (see (25.31)).

Multiplying the heat inequality (25.12) by 'ljJ^2 vP and integrating by parts

in space and time, we have.

(25.15)

0 2: 1T

2

r 'lj;^2 (~ ~ ( v^2 P) - vP ~ ( vP) + p ( Q + A) v^2 P) dμ dT

Tl lv 2 Eh.

. = 1

72

{ (-'l/J EJ.'l/J v^2 P - ~'lj;^2 v^2 PR) dμ.dT + ~ ( { 'lj;^2 v^2 Pdμ) (T2) '

Tl lv OT 2 2 lv

+ j"T2 r (IV' ( 'lj;vP) 12 - v2p IY''l/Jl2 + p ( Q +A) 'l/;2v2p) dμ dT

Tl JD

since the volume form evolves by gT dμ = Rdμ and since

• fv 'l/;2vP ~ (vP) dμ = k IY' ('l/;vP)l2 dμ -iv v2P IY''l/Jl2 dμ.

Note that, in retrospect, we were able to throw away the good_ gradient
term on the RHS of (25.12) since, in (25.15), integrating by parts on the
term involving-~ (vP) yields a comparable good term. ·
Now choose A E [O, oo), depending only on Q and SUPnx[o,T] R (in par-
ticular, A is independent of p and D), so that

1

(25.16) Q +A -

2

P R 2: O

on D x [O, T] for all p 2: 1. Applying this to (25.15), we have

(25.17) 02: 1:

2

fv1\7('lj;vP)l^2 dμdT+~ (fv 'l/;^2 v^2 Pdμ) (T2)

-1:

2

fv ('l/J~~ + IY''l/J1

2

) v

2

PdμdT.

Thus, given 0 :S Ti < T2 :ST, if 'ljJ : n x [ri, T2] -+ [O, 1] in (25.13) with

'I/; (·,Ti) = 0 is defined so that it further satisfies the inequality

8'1/; 2

(25.18) 'ljJ OT + IY''l/Jlg(T) :SL

for some constant L E [O, oo), then since· both terms on the RHS of the first
line of (25.17) are nonnegative, we have