194 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE
The result follows when we note that 1Rc l^2 :::; 1Rml^2 , discard the negative
terms on the second line, and compute
p ~ A2 (μ v)2 + μ2 (>- v)2 + v2 (>-_ μ)2
= 2 [>-2μ2 + A2v2 + μ2v2 _ (>-2μv + μ2>-v + v2>-μ)J.
0
- The gradient estimate for the scalar curvature
In this section we obtain a gradient estimate for the scalar curvature.
This estimate is important because it enables us to compare curvatures at
different points, whereas the pinching estimate of Section 5 is a pointwise
estimate which compares curvatures at the same point.
As motivation, let us recall Schur's Lemma. If g is an Einstein metric
on a manifold Mn, then Re = f · g for some function f on Mn and one has
V'kR = V'k (gij Rj) = n\7kf.
On the other hand, the contracted second Bianchi identity implies that\7 kR = 2\7j Rjk = 2\7j (J gjk) = 2\7 kf.
So we have (n - 2) \7 f =: 0, which shows that f = R / n is constant in
dimensions n > 2.
Now suppose we have a solution ( M^3 , g ( t)) of the Ricci flow on a closed
3-manifold. The pinching estimate (6.37) proved in Section 5 can be written
in the form
(6.41) IRc-lRgl2 -
~-~ 3 --< C R-o
R2 - 'where the left-hand side is a scale-invariant quantity which measures how
far the metric is from being Einstein, and the right-hand side is small when
the scalar curvature is large. So if the scalar curvature becomes uniformly
large, it is natural to expect its gradient to approach zero uniformly. It is
also natural to expect that the contracted second Bianchi identity will be
key to proving this result. Both expectations are in fact correct, and we
shall now establish the following result.
THEOREM 6.35. Let (M^3 , g (t)) be a solution of the Ricci flow on a
closed 3-manifold with g (0) = go. If Re (go) > 0, then there exist "jj, J > 0
depending only on go such that for any /3 E [ 0, "iJ], there exists C depending
only on /3 and go such that----W:-l\7Rl2 :::; /3R-8/2 + C R-3.
Here, the left-hand side is a scale invariant quantity, while the right-hand
side is small when the scalar curvature is large ..