1547671870-The_Ricci_Flow__Chow

5. LOCAL PINCHING ESTIMATES 191
   0 0
   where as in (6.34), Rm and Re are the trace-free parts of the Riemann
   curvature operator and Ricci tensor, respectively. Once again, the left-hand
   side is scale invariant, while the right-hand side tends to 0 uniformly if
   infxEM3 R (x , t) ----+ oo.

We shall give two proofs of the theorem: the first uses the maximum

principle for systems and is in the spirit of [59], whereas the second is closer

in spirit to the original argument of [58].

FIRST PROOF OF THEOREM 6.30. Because

d

dt (). - v) = ( ,\ - v) ( ,\ - μ + v) '

we may assume that ,\(M) > v(M) for M(t) E (/\^2 T*M^3 @s/\^2 T*M^3 )x.

We calculate

and

Hence

d

• log ( ,\ - v) = ,\ - μ + v
  dt

d v2 + μ2

-d log ( v + μ) = ,\ +

t v+μ

d

d log [ ,\ - ~ -J l = ( ,\ - μ + v) - ( 1 - c5) (,\ + v2 + μ2)

t (v+μ) v+μ

v2 + μ2

= c5,\ + (v - μ) - (1 - c5) --

v + μ

1

:::; c5,\ -

2

(1 - c5) (v + μ),

where we used v :::; μand v^2 + μ^2 :2: ~ (v + μ)^2 to obtain the last inequality.

By Lemma 6.28, the right-hand side is nonpositive for c5 chosen small enough.

Now define the convex set

JC= { lP': [,\(JP') - v (JP')] - C [v (JP') + μ (lP')]^1 -^0 :::; O}

and continue as before. D

SECOND PROOF OF THEOREM 6 .30. By Lemma 6.34 (below), the func-

ti on 0

f

= IRml2

. R2-E.
satisfies the evolution equation

(6.38) ~~ :::; .6.f + 2 (1 - c) (V' f , Y' (log R)) + 2Q.

Here

Q ::!;= R;-E. ( c 1Rml^2 I Rml^2 - P) ,