1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

l. REDUCED DISTANCE OF TYPE A SOLUTIONS 143

independent of i. Using the Banach- Alaoglu theorem, there exists a subsequence

such that iii converges weakly in W^1 •^2 (K x [t 1 , t 2 ]) to f=l/Cx(t t 1.^3 Hence

/Cx[ti.t 2 ]^1 • 2

(30.37) ( v ei, V <p) _ - --;:;-'P dμ 9 , dt

i

t

2

j ( _ eii )

t 1 Moo 9, ut

i

t2 r ( ee= )

---+ t1 j Moo (V £=' V <p) 900 - fit tp dμ9oo dt.

STEP 3. Convergence of ftt,^2 J Moo [Vii[~, <p dμ9, dt. We divide the proof into

two substeps.

STEP 3A. Lower bound for lim infi-t= ftt,^2 J Moo f Vii[~, <p dμ 9 , dt. We have

0 ~ i t

2

r lve{X) - vii /

2

'P dμ9oodt

t 1 j Moo 900

= it2 1 Ive= I ~00 'P dμ9oo dt + it2 1 Iv ii I ~00 'P dμ9oo dt

t1 Moo t1 Moo

• 2 jt

2

j ( v e{X)' vii) 'P dμ9oo dt.

t 1 M 00 900

Taking the lim inf of this implies that

(30.38) li~~fit

2

1 [Vii[~ 00 <pdμ9 00 dt ~ it

2

1 [V£{X)[~ 00 <pdμ9 00 dt.

t1 Moo t1 Moo

By the convergence of gi to g=, we obtain

lim inf i t

2

f Iv ii I~' 'P dμ 9 , dt ~ it

2

j Ive= I ~ 00 'P dμ 900 dt.

t-t<Xl t1 } Moo t1 Moo

STEP 3B. Upper bound for limsupi-+= ftt,^2 JM 00 [Vii[~ydμ9,dt. Since ii con-

verges weakly to e{X) in W^1 •^2 '

(30.39) limsuplt

2

j 1viif~ootpdμ9oodt-it

2

r [Vf{X)l~oo tpdμ9oodt

t-t<Xl t1 Moo ti } Moo

= lim sup it

2

j (vii - v e{X)' 'P vii) dμ9oo dt.

i-+= t 1 M 00 900

By Cheeger- Gromov convergence, we may replace g= by gi in the LHS of (30.39).
Step 38 shall follow from showing that

(30.40)

is nonpositive.
The argument below of Enders follows Lemma 9.21 in Morgan and Tian [251].
Recall from (7.92) in Part I that for each i sufficiently large we have the elliptic
partial differential inequality:

• 
  
  
  
  • 2 ii-n
    2t::.. 9 ,ei - 1vei1 9 , + R9, + wi _ t ~ o,
  
  
  
  

3The choice of metric on M= x (a=, w=) used to define W^1 •^2 (K x [t1, t2]) is not essential.