1547671870-The_Ricci_Flow__Chow

260 9. TYPE I SINGULARITIES

PROOF. By (6.32), we calculate

v^2 dt d D (M) = -v dt d (tr M) + (tr M) dt d v - v dv dt

d d

= -v dt (>. + μ + v) + (>. + μ) dt v

= -1/3 - I/ (>.2 + μ2 + >.μ) + (>. + μ) >.μ.

Thus it suffices to prove that

7f ~ -1/ (>.2 + μ2 + >.μ) + (>. + μ) >.μ 2: 0

whenever v < 0. There are two cases: ifμ< 0, then we write

7f = (μ v) (>.2 + μ2 + >.μ) μ3 2: -μ3 > O,

while if μ 2: 0, we have

7f = >.^2 (μ - v) - vμ^2 + >.μ (μ-v) 2: 0.

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PROOF OF THEOREM 9.4. Fix any x E M^3 with v (x, 0) < 0. Let M be

the quadratic form corresponding to Rm [g], and let

/Cc (A^2 TM^3 &Jsf\^2 TM^3 )x

be the time-dependent set defined in Lemma 9.5. Then since

JC (O) = {JP I tr JP 2: -3 and }

trJP2: lv(JP)I (logjv(JP)j-3) if v(JP)::; -1 '

we have M (0) E JC by the hypothesis inf v (-, 0) 2: -1. We claim M (t)

remains in JC as long as v < 0. The theorem follows directly from the claim,

which implies that

=-R -log ( -v) 2: -i+3t (1)

I/ l+t^1 - - log -- 1 + t = log (^1 + t) -^3

when 0 < -v < 1/ (1 + t), and

R

• 2: log (- v) +log (1 + t) - 3
  
  
  • 1/
    when - v 2: 1/ (1 + t).
    The proof that M (t) E JC uses the maximum principle for systems. The
    inequality tr M 2: -3/ (1 + t) is preserved, because
    d 1 1
    dt (trM) - 3 (trM)2 2: 3 (>.2 + μ2 + v2) 2: 0.
  
  
  
  

And if lvl 2: 1/ (1 + t), then Lemma 9.6 implies that ftn (M) 2: 1/ (1 + t),

which is equivalent to the inequality

:t [ tlv7 - log lvl - log (1 + t)] 2: 0.

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