- NO FINITE TIME LOCAL COLLAPSING 263
k. We can apply Remark 6.66 to the ball B (p, r /2k) and get Vol B (p, r /2k) 2:
"'o (r /2kt. Hence
VolB (p, r) > 3nvolB (p, r/2)> 3n(k-l) Vol B (p, r /2k)
2: 3n(k-l) "'O (r /2k) n> "'o rn.- 2n
Hence the proposition holds with c (n, T, C 1 ) = co(n2;,c^1 ), since by Remark
6.66 we have "'O =co (n, T, C1) e-A. D5.3. Application of K,-noncollapsing to the analysis of singular-
ities. We present some applications to end this section. In the next section
we give an improvement of the no local collapsing theorem.
5.3.1. Existence of finite time singularity models. An injectivity radius
estimate (Corollary 6.62) implies that one can apply Hamilton's Cheeger-
Gromov-type compactness theorem to obtain the existence of singularity
models for singular solutions with finite singularity time. (Recall that the
definition of a singularity model is given in Remark 1.29.)
THEOREM 6.68 (Existence of singularity models). Let g(t), t E [O, T), bea smooth solution to the Ricci flow on a closed manifold Mn with T < oo.
Suppose that there exists a sequence of times ti /' T, points Pi EM, and a
constant C < oo such that
(6.92)
(6.93)Ki ~ I Rm g(ti) (Pi) I --+ oo,
IRmg(t)(x)I :s;CKi forallxEM andt<ti.Then there exists a subsequence of the sequence of dilated solutions^18gi ( t) ~ Ki · g (ti + ;J
such that (Mn, gi(t),Pi) converges to a complete ancient solution to the
Ricci flow (M~, g 00 (t),p 00 ) in the sense of C^00 -Cheeger-Gromov conver-
gence. Furthermore there exists "'> 0 such that g 00 (t) is K,-noncollapsed on
all scales.
PROOF. By Perelman's no local collapsing theorem, we have an injec-
tivity radius estimate at the points Pk with respect to the metrics gk (0).
Hence by Hamilton's compactness theorem (Theorem 3.10), there exists a
subsequence such that (Mn,gi(t),Pi) converges to a complete ancient so-
lution (M~, g 00 (t),p 00 ) to the Ricci flow. Since g (t) is K,-noncollapsed on
the scale VT for all t E [O, T), we have gi(t) is K,-noncollapsed on the scale
18 As usual we denote a subsequence of i still by i rather than ij to simplify our
notation.