1547671870-The_Ricci_Flow__Chow

240 8. DILATIONS OF SINGULARITIES

DEFINITION 8.16. A solution (Mn, g(t)) of the Ricci fl.ow on a time

interval [O, T) is K;-noncollapsed on the scale p if for every metric ball

Bg(t) (x 0 , r) of radius r < pin which the curvature is bounded from above

in the sense that

sup{jRm(x,t)I :xEBg(t)(xo,r), O::St<T} :::;r-^2 ,

one has a lower bound on volume of the type

Vol [Bg(t) (xo,r)] 2:: K;rn.

If (Mn,g(t)) obeys a suitable injectivity radius estimate, the Com-
pactness Theorem from Section 3 of Chapter 7 shows that a subsequence
of the pointed sequence (Mn, gi ( t), Xi) converges to a complete pointed
solution (M~, g 00 (t), x 00 ) of the Ricci fl.ow on an ancient time interval

-oo < t < w :::; oo. This singularity model satisfies

jRm [goo] (x, t) j 900 ::SC

for all (x, t) EM~ x (-oo, O].
If (Mn, g(t)) becomes singular in finite time - in other words, whenever
(Mn,g(t)) encounters a Type I or Type Ila singularity - Perelman's No
Local Collapsing Theorem [105] provides such an estimate. In particular,

Perelman's result implies that there exists "' > 0 such that the singularity

model (M~, g 00 (t), x 00 ) is K;-noncollapsed at all scales.

THEOREM 8.17 (Perelman). Let (Mn,g (t)) be a solution of the Ricci

flow that encounters a singularity at some time T < oo. Then there exists

a constant c > 0 independent oft and a subsequence (xi, ti) such that

c

inj(xi,ti) > ~=======

• JmaxMn jRm (-, ti)I

In forthcoming work, we plan to provide a detailed explanation of the

proof.

REMARK 8.18. A global injectivity radius estimate on the scale of the
maximum curvature was originally established for Type I singularities by
Hamilton's isoperimetric estimate. (See §23 of [63].)

4. Dilations of finite-time singularities

There is a lower bound for the curvature blow-up rate which applies
both to Type I and Type Ila singularities.

LEMMA 8.19. Let Mn be a compact manifold. If 0:::; t < T < oo is the

maximal interval of existence of (Mn, g (t)), there exists a constant c 0 > 0

depending only on n such that

(8.10) sup !Rm (x, t) I 2:: Teo.

xEMn - t