- THE COMPACTNESS OF A;-SOLUTIONS 139
REMARK 20.13 (Heuristic). A priori, it is conceivable that it is pos-
sible for a sequence of pointed t;;-solutions (Mk,gk (t) ,xk), t E (-oo,O],
to the Ricci fl.ow to have at t = 0 uniformly bounded curvature at the
basepoints Xk but also to have curvature blowing up at points Yk with
d 9 k(o) (Yk, xk) ::; C. For example, one can imagine a 'conical-type sin-
gularity' forming near the points Yki that is, a Gromov-Hausdorff limit
(M~,doo (t) ,xoo) of (Mk,9k (t) ,xk) ask-+ oo may exist which is a Eu-
clidean metric cone based at Yoo and such that M 00 - Yoo is a smooth
n-manifold. Note that for such a limit, were it to exist, we would have
R 00 (z) d'?xi (z, y 00 ) ::; const < oo. Needless to say, by the above compactness
theorem, this cannot occur for n-dimensional t;;-solutions with Harnack.
We shall give two proofs of Theorem 20.9.3.2. First proof of the compactness of t;;-solutions - via Propo-
sition 20.4.
In this subsection we give a proof of the compactness Theorem 20.9 using
Proposition 20.4. This is the proof which Perelman gave in §11.7 of [152].
Let {(Mk,gk (t))}~ 1 be a sequence in 911:~,a;n, Xk E Mk and tk ::; 0.
Let {gk (t)} be the curvature normalized sequence of solutions defined by
(20.21):
Let^15
Ak ~ sup R_gk(z,O)d~k(o)(xk,z)
zEMk
= sup R 9 k (z, tk)d;k(tk) (xk, z).
zEMkNote that if Mk is noncompact, then
Ak 2: ASCR (gk (0)) = ASCR (gk (tk)),
where the asymptotic scalar curvature ratio ASCR is defined by (19.8). We
first choose a subsequence {(Mk,9k (t))} such that either
(a) Ak::; 1 for all k or(b) Ak > 1 for all k.
We shall extract a convergent subsequence in 9J1:~8;n in either (a) or (b).
One of the main ideas in the proof of this theor~m is to obtain uniform
curvature bounds so that one can simply apply Hamilton's (local) Cheeger-
Gromov-type compactness theorem.
Case (a). Ak ::; 1 for all k. In this case we have for each k for which
Mk is noncompact that
ASCR (gk(O)) ::; 1.(^15) More generally, given a pointed Riemannian manifold (Mn,g,O), we may define
the 'maximum scalar curvature ratio' as A (g) ~ supxEM R (x) d^2 (x, 0).