1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. PROPERTIES OF SINGULARITY MODELS 4 1

oo such that if we set gi(t) ~ K ig(ti+Ki-^1 t), then (Mn,gi(t),(xi ,O)) con-

verges in the C^00 pointed Cheeger- Gromov sense to a nonfiat complete so lution

(M~, 900 (t), (x 00 , 0)), t E (-oo, OJ, with uniformly bounded sectional curvature.

A consequence of Theorem 28.6 is that any singularity model is 11:-noncollapsed

below all scales for some 11: > 0. This is true because b eing 11:-noncoll apsed b elow a

fixed scale is preserved under Cheeger- Gromov limits and "b elow all scales" follows

from the rescaling factors tending to infinity.

THEOREM 28.9 (Singularity models are 11:-noncoll apsed below all scales). Let

(Mn, g( t)), t E [O, T), be a finite-time singular so lution on a closed manifold. Then

there exists 11: = 11: (n, g(O), T) > 0 with the following property. For any associated

singularity model (M~, g 00 (t)), t E (-oo, O], and for any x 0 E M 00 , t 0 E (-oo, O],

and ro > 0 such that the scalar curvature satisfies R 900 (to) :::; r 02 in B 900 (to) (xo, ro),

we have

Note that we may choose the constant 11: (n, g(O), T , p) in Theorem 28.6 so that

it is nonincreasing in p. We may then choose 11: (n, g(O), T) in Theo rem 28.9 to b e

equal to limp--+D 11: (n , g(O), T , p).

REMARK 28.10. The above discussion extends , with modifications, to complete
solutions of the Ricc i flow on noncompact manifolds (using Theorem 8.26 in Part I).

When the underlying ma nifold M 00 of a singularity model is compact, it is
diffeomorphic to the underlying manifold M of the finite-time singula r solution
from which it arises. Rema rkably, Z.-L. Zha ng has classified compact singularity
models in the following sense.

THEOREM 28 .11 (Compact singularity models). Any compact singularity model

is a shrinking G RS.

EXERCISE 28.12. Show that a compact singularity model cannot b e Ricci flat.

We h ave the following consequence of no local coll apsing regarding the volume
growth of noncompact singularity models.

COROLLARY 28.13 (Linear volume growth if R :::; C). If (M~ 1 g 00 (t)) is a

noncompact singularity model and if t is such that the scalar curvature R 900 (t) is

bounded from above, then for any 0 E M 00 the volume of B 900 (t) (O,r) grows at

least linearly in r.

PROOF. By Theorem 28.9, if R 900 (t):::; Con M 00 , then for any Xo E M 00 ,

for some 11: > 0. Since g 00 (t) is co mplet e and M 00 is noncompact, there exists

a unit sp ee d ray I ema nating fr om 0. The corollary follows from summing this