1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

15. The quantity Q must be identically zero

and

(29.199)

8

at l~eucvil

2

= V~euc l~eucvil

2

• 2v l~eucval
  2
  
  
  • 4~eucV l~eucvil
    2
  
  
  
  

+ 2vi~eucVi~~uc V - 8~eucViV£jiVej.

From (29.198), (29.199), and dropping the vanishing terms

~eucv(lvijk/

2

• l~eucvi) - 2V£iVijkV£jk + 2VjkVijk~eucVi = 0,
  we compute using (29.175) that

(29.200)

8

at laijk 1

2

= V~euc laijk 1

2

• 4R laijk /

2

2

• ::(l8e (vi.iijk)l
  2
  
  
  • l"Vv l
    2
    li.iijkl
    2
    
    
    • 3vi.iejkVe~eucVjk)·
      v
    
    
    
    
  
  
  
  

To analyze the RHS of (29.200), let Zjk = ~eucVjk - ~~;ucVbjk· Since

I--^1

(^2 1) l"-1 (^2) 1- 12

veaejk = - v v aijk

2

125

(which is analogous to (29.161)) and since 15ei8e (vaijk) = vei.iejk + ~Zjk, by (29.157)

we have

/Be (vaijk)l^2 - I TF(8 (va))eijkl^2 = l8i (vaijk)l^2

1 -2

= 2 l"Vvl

2

laijkl

2

+ viaijkVZjk + v

4

lzjkl^2.

Applying

2 l"Vvl

2

laijkl

2

+ vvt /aijkl; = 2veaijk TF(a(va))eijk + 2 lvtiitjkl

2

+ i!ti.itjkVZjk

to (29.200), we calculate t hat

BQ - - -

Dt = V~eucQ - 4RQ

- 2 (lae (vaijk)l

2

+ ~ l"Vv l

2

li.iijkl

2

• 3vei.iejkV~eucVjk + i!i!t laijkl;)

= v~eucCJ - 4RQ - 2(1 TF(a(va))eijk + i!ti.iijkl^2 - li!tiitjkl

2

)

• 2


• 21 ~zjk - vei.iejk I

Again applying (29.157) yields (29.196). This completes the proof of Lemma 29.48.

15. The quantity Q must be identically zero

Let Q = v /TF(B)l^2 , as defined in (29.134). In this section we shall prove the

following. Since we have shown in Proposition 29.46 that Q being zero implies that
g(t) is either a shrinking round 2-sphere or the King- Rosenau solution, this will
complete the proof of the main Theorem 29.l.

PROPOSITION 29.49. Ifμ> 0 in Proposition 29.36, then Qmax(t) ~ maxx Q(x, t)
is nonincreasing and satisfies

(29.201) lim Qmax(t) = 0.

t-+-oo

Hence Q(x, t) = 0 on S^2 x (-oo, 0).