1547671870-The_Ricci_Flow__Chow

10. EXPONENTIAL CONVERGENCE 219

In particular, the eigenvalues>. ~ μ ~ v of M (which are twice the sectional
curvatures) evolve by

It is easy to check that

d •

• 
  

  
  

  . = >.^2 + μv - r >.
  dt
  d 2

  
  

  
  

dtμ=μ +>.v- rμ

d 2

dt v = v + >.μ - rv.

d >.

[>. (μ + v)] -d log --= (μ + v) (>.^2 + μv) - r>. (μ + v)

t μ+ v

• 
  

  
  

  . [μ^2 + v^2 + >. (μ + v) J + r >. (μ + v) ,

  
  

  
  

hence that the proof of Lemma 6.28 goes through for the normalized flow
exactly as written. Thus we can immediately state the following result.

LEMMA 6.65. Let (M^3 ,g(t)) be a solution of the normalized Ricci flow

on a closed 3-manifold of initially positive Ricci curvature. Then there exists

a constant B such that

>.:SB(μ+ v)

for all positive time.

REMARK 6.66. One can also conclude directly that the estimate of
Lemma 6.28 applies to the normalized Ricci flow by writing it in the scale-
invariant form

.
--μ +v-<B.

We shall need one more simple observation.

LEMMA 6.67. Let (M^3 ,g(t)) be a solution of the normalized Ricci flow

on a closed 3-manifold of initially positive Ricci curvature. Then there exists

c: > 0 depending only on go such that R ~ c: for all positive time.

PROOF. By (6.6), R satisfies the differential inequality 8R/8t ~ b..R.

Hence Rmin (t) ~ Rmin (0) > 0. D

Now we are ready for the key estimate of this section.

PROPOSITION 6.68. Let (M^3 , g(t)) be a solution of the normalized Ricci

flow on a closed 3-manifold of initially positive Ricci curvature. Then there

exist constants a E (0, 1), /3, and C depending only on go such that

. - v :S C (μ + v)^1 -°' e-f3t

for all positive time.