1547671870-The_Ricci_Flow__Chow

218 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

( 4) The Ricci tensor of g evolves by

a e

at Rjk = b..LRjk = b..Rjk + 2RpjkqRpq - 2RjR£k·

(5) The scalar curvature R of g evolves by

a 2

atR = b..R + 2 IRcl - rR.

10. Exponential convergence

We may now assume we are given a solution (M^3 , g ( t)) of the normal-

ized Ricci flow

a

atg = -2 Re +rg

starting on a closed 3-manifold of positive Ricci curvature, where r (t) is
given by formula (6.61). By Theorem 6.58, we know that g (t) exists for all
positive time. By Lemma 6.8, Corollary 6.11, and Lemma 6.57, we know
that the Ricci and scalar curvatures of g (t) remain strictly positive. By
Lemma 6.60, we know that there exists a positive constant A such that

(6.62) O<R::;A,

hence such that

(6.63) 0 < r:::; AV,

where V is the constant volume of (M^3 , g (t)). And by Corollary 6.58, we

know g (t) becomes asymptotically Einstein in the sense that

Jim ( sup l~cp) = o.

t-+oo xEM3 R

In order to show that this convergence is exponential, we shall follow

the method of Section 4 (which uses techniques from the later paper [59]

instead of the method originally employed in [58]). As in Section 2, let

V be a vector bundle over Mn isomorphic to T Mn. Corresponding to the

evolution of g (t) by the normalized Ricci flow, we construct a one-parameter

family of bundle isometries 1, ( t) : V -t T Mn evolving by

a r

- 1, = Rcoi - -1,.

at 2

Then recalling Corollary 6.64, we compute that the analogue of formula

(6.21) for the evolution of the components (Rabcd) of 1,* Rm is

a

at Rabcd = b.. DRabcd + 2 ( Babcd - Babdc + Bacbd - Badbc) - r Rabcd.

It follows that the quadratic form M defined by the curvature operator on

each fiber of the bundle /\^2 T M^3 evolves by the ODE

d 2 #

dt M = M + M - r M.