190 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE
Let >. (lfD) 2: μ (lfD) 2: v (lfD) denote the eigenvalues of
lfD E (A^2 T M^3 0s A^2 T M^3 )x.
Define JC c (A^2 TM^3 0s A^2 TM^3 )x by
JC ~ {lfD : >. (lfD) - C (v (lfD) + μ (lfD)) ::; O}.
It is easy to see that JC is invariant under parallel translation. JC is convex
in each fiber, because the function
. (lfD) - C (v (lfD) + μ (lfD)) = max lfD (U, U)+ C max [-lfD (V, V) - lfD (W, W)]
IUl=l IVl=IWl=l
(V,W)=O
is convex. Let M (t) E (A^2 T M^3 0s A^2 T M^3 )x be the quadratic form
corresponding to Rm [9 (t)], and choose C < oo so large that M (0) E JC for
all x E M^3. By (6.36), M remains in JC. D
This estimate implies a global bound which will be useful later.COROLLARY 6.29. Let (M^3 , 9(t)) be a solution of the Ricci flow on a
closed 3-manifold of initially positive Ricci curvature, and define Rmin (t) ~
infxEM3 R (x, t). Then there exists /3 > 0 depending only on 9o such that at
all points of M^3 ,
Re 2: 2/3^2 R9 2: 2/3^2 Rmin9 ·
PROOF. By the lemma and formula (6.33), there exists C > 0 depending
only on 90 such that the inequality
Rμ+v A >.+μ+v R Rmin
c -> -- 2 9 - > 2C - (^9) - > 6C (^9) - - 6C - 9 > - --6C 9
holds everywhere on M^3. D
Lemma 6.28 also helps us prove the following result, which shows that
the metric is nearly Einstein at points where the scalar curvature is large.
THEOREM 6.30 (Ricci pinching is improved.). Let (M^3 ,9(t)) be a solu-
tion of the Ricci flow on a closed 3-manifold such that 9o has strictly positive
Ricci curvature. Then there exist positive constants J < 1 and C depending
only on 9o such that>.- v C
--<--~
v + μ - (v+μ)8.
In particular, the left-hand side is invariant under homotheties of the metric,
while the right-hand side tends to 0 as v + μ---+ oo.REMARK 6.31. Theorem 6.30 is equivalent to the following statement,
which was proved in Hamilton's original paper [58]: There exist constantsJ > 0 and C < oo depending only on 9o such that
0 0
(6.37) 1Rml(^2) _ 1 Rcl (^2) < - 8
R2 - 4 R2 - CR ,