1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. PROPERTIES OF SINGULARITY MODELS 45

THEOREM 28.24 (Existence of asymptotic shrinker). Let (Mn, g ( T)), T E

[O, oo), be a r;,-solution to th e backward Ricci flow and let p E M. For any se-

quences Ti -t oo and Qi E M such that the reduced distance based at (p , 0) satisfies

the estimate

£(p,O) (qi, Ti) :S C

for some constant C independent of i (note that for any Ti such Qi exist with C = ~),

there exists a subsequence such that (Mn, Ti-^1 g (TiT), (qi, 1)) converges to a com-

plete nonfiat r;,-noncollapsed shrinking gradient Ricci so liton (M~,g 00 (T), (q 00 , 1))

with nonnegative curvature operator. Moreover, if n = 3, then the curvature of

goo ( T) is bounded.

R egarding the geometric properties of singularity models, some in relation to

GRS, we h ave the following questions. Recall that an a ncient solution (Mn,g(t)),

t E (-oo, 0), is called Type I if there exists C < oo such that IRml (x, t) :::; 1 ~ on

M x ( -oo, -1 J. Otherwise the ancient solution is called Type II.

PROBLEM 28.25 (Singularity models which are not solitons). Does there exist

a noncompact Type I singularity model which is not a shrinker? Does there exist

a noncompact Type II singularity model which is not a steady?

PROBLEM 28.26 (Can a singularity model b e Ricci flat?). Can a noncompact

singularity model be Ricci flat? Hopefully the answer is no (by Exercise 28.12 and

Lemma 28. 16 , this would imply Optimistic Conjecture 28.15).

R egarding the volume growth of singularity models, note the following.

EXAMPLE 28.27 (Volume growth of the Bryant soliton). The n -dimensional

Bryant soliton ( n ~ 3) satisfies

Vol B (p, r) (O )

}~~ r(n+l)/2 E ' 00.

On the other hand, for the cylinder sn-^1 xlR, where sn-^1 is the unit (n - 1)-sphere,

the volumes of balls grow linearly.

Because of this example, qualitatively speaking, the following is the best one
can hope for.

OPTIMISTIC CONJECTURE 28.28 (Lower bound for volume growth of nonsplit-
ting singularity models). The volume growth of any n-dimensional noncompact
singularity model, provided its universal cover is not isometric to the product of JR
with an (n - 1)-dimensional solution, is at least of the order r(n+l)/^2.

The b est possible upper bound for the volume growth of singularity models is
less clear. One may start with this (compare with Theorem 27.42):

OPTIMISTIC CONJECTURE 28. 29 (Upper bound for volume growth of singu-
larity models). The volume growth of any n-dimensional noncompact singularity
model is at most of the order rn.