144 3. THE COMPACTNESS THEOREM FOR RICCI FLOWPROOF. Let (M, g (t)), t E [O, T), be the maximal solution to the Riccifl.ow with g (0) = g 0. Recall that TS: ~ (Rmin (o)r^1 < oo. The proof relies
on three main estimates.
(1) (Positivity of Ricci is preserved.) Re (g (t)) > 0 for all t ~ 0.
(2) (Strong curvature pinching.) There exist C < oo and 6 > 0 depend-
ing only on go such that[Rc-~Rg[ < CR-o
R -
on M x [O, T) (see inequality (6.37) on p. 190 of Volume One).(3) (Jnjectivity radius estimate.) There exists LO > 0 depending on
n, T, go such that if (x, t) EM x [O, T) and r E (0, 1] are such that
IRm(·,t)I S: r-^2 in Bg(t) (x,r),
then injg(t) (x) ~ lor (see Theorem 3.28 or Corollary 6.62).
The main technique is to dilate and apply the compactness theorem.
Choose (xki tk) with Xk EM and tk ~ T such that
Rk ~ R (xk, tk) = max R ~ max IRml ~ oo
M3 x [O,tk] M^3 x [O,tk]
ask~ oo. Consider the sequence of solutions (M, gk (t), xk), t E (-tkRk, O],
where
gk(t) ~ Rkg (tk + R"k^1 t).
Let rk ~ R~^1 /^2 , which is bounded above by 1 for k large enough. We have!Rm (·,tk) I S: r"k^2 in Bg(tk) (xk, rk).
Thus by (3), injg(tk) (xk) ~ lork, which is equivalent to inj 9 k(o) (xk) ~ Lo.
Hence Hamilton's compactness theorem implies that, for a subsequence,
(M, gk (t), xk) converges to (M~, g 00 (t), x 00 ), a complete solution definedfort E (-oo,O]. We claim that (M 00 ,g 00 (t)) has constant positive sectional
curvature. In particular, M 00 has bounded diameter and hence is diffeo-morphic to M. So the theorem follows.
First note that R 900 (xoo, 0) = limk--+oo R 9 k (xk, 0) = limk--+oo 1 = 1.
Hence R 900 (x, 0) > 0 for x contained in a neighborhood of x 00 • Estimate (2)
says that[Re (gk) - ~R (gk) gk[ (t) < CR-o R ( )-o
R (gk) - k gk ·
Since the convergence of (M, gk (t), xk) is in C^00 on compact subsets, wehave on the subset of M 00 where R (g 00 ) > O,
~~~~~~~---'-~-'---~~~--=--~~~~ [Rc(gk)-~R(gk)gk[ [Rc(goo)-~R(goo)goo[
R (gk) R (goo)
(we have swept under the rug the fact that in Cheeger-Gromov convergence
one must pull back by appropriate diffeomorphisms from an exhaustion of
Moo to M; we leave it to the reader to justify the arguments in this proof).