174 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE
of the unnormalized flow modulo rescaling; but the advantage of finally
converting to the normalized flow is that it will allow us more naturally to
demonstrate exponential convergence. Our strategy for proving Theorem 6.3
will be to learn enough about how curvature evolves under the unnormalized
flow to obtain certain a priori estimates that will suffice to prove long-time
existence and convergence of its normalized cousin.
3. The structure of the curvature evolution equation
In this section, we compute the evolution equations for the Levi-Civita
connection and the Riemann, Ricci, and scalar curvatures of a solution to
the Ricci flow. We start by recalling certain general evolution equations
for a one-parameter family of metrics which were derived in Section 1 of
Chapter 3.
REMARK 6.4. Unless explicitly stated otherwise, all results in this section
hold for any dimension n 2: 2.
LEMMA 6.5. Suppose that 9 (t) is a smooth one-parameter family of met-
rics on a manifold Mn such that
[)
8t9 = h.
(1) The Levi-Civita connection r of 9 evolves by
[) k 1 kf.
f)t rij = 29 (''Jihje + 'Jjhif. - \Jehij).(2) The (3, l)-Riemann curvature tensor Rm of 9 evolves by8 e 1 £p { 'Ji'Jjhkp + 'Ji'Jkhjp - 'Ji\Jphjk }
8t Rijk = 29
-'Jj\Jihkp - 'Jj'Jkhip + \Jj\Jphik
1 { 'Ji'Jkhjp + \Jj\Jphik - 'Ji\Jphjk - 'Jj\Jkhip }
=2~.
-R{jkhqp - R{jphkq
(3) The Ricci tensor Re of 9 evolves by
[) 1
f)tRjk = 29pq ('Jq\Jjhkp + 'lq\Jkhjp - \Jq\Jphjk - \Jj\Jkhqp)= -~ [ ~Lhjk + 'lj\Jk (tr 9 h) + 'lj (Shh+ 'Jk (8h)j] ,
where ~L denotes the Lichnerowicz Laplacian defined in formula
{3.6).
(4) The scalar curvature R of 9 evolves by
a .. ke
atR = 9iJ9 (-'Ji'Jjhkz + 'Ji'Jkhje-hikRje)