1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. PROPERTIES OF SINGULARITY MODELS 43

OPTIMISTIC CONJECTURE 28 .18. The asymptotic cone of a noncompact sin-

gularity model at any fixed time is well defined (i.e., unique).

This conjecture is true in dimension 3 (see Theorem 28.30 below).

2.2. Singularity models and 11:-solutions in dimension 3.

Singularity models in dimension 3 are particularly nice. For our discussion

of these objects, we recall the following class of ancient solutions introduced by

Perelman.

DEFINITION 28.19 (11:-solution). Given a positive constant 11:, a complete ancient

solution (Mn,g(t)), t E (-oo, O], of the Ricci flow is called a11:-solution ifit satisfies

the following:

(i) For each t E (-oo, OJ the metric g(t) is nonfiat with nonnegative curvature

operator and is 11:-noncollapsed at all scales.

(ii) There is a constant C < oo such that t he scalar curvature R 9 (x, t) :::::: C

for all (x, t) EM x (-oo, OJ.

The work of Perelman implies the following, which is equivalent to saying that

Optimistic Conjecture 28.14 is true when n = 3 (see [76]).

THEOREM 28.20 (3-dimensional singularity models must have bounded curva-

ture). Any 3-dimensional singularity model must be a 11:-solution.

To prove this theorem, we shall use the following canonical neighborhood

theorem of Perelman.

THEOREM 28.21. For any r:: > 0, p > 0, and 11: > 0, there exists r 0 E (0, lJ with

the following property. Let (M^3 , g (t)), t E [O, T) with T E (1, oo ), be a solution to

the Ricci flow on a closed 3 -manifold which is 11:-noncollapsed below the scale p and

suppose that (x 0 , t 0 ) is such that t 0 E [1, T) and Q ~ R (x 0 , t 0 ) 2:: r 02. Then the

solution

g (t) ~ Qg (to+ Q-^1 t) on B 9 (o)(xo, r::-^1 /^2 ) x [-r::-^1 , OJ

is r::-close in the cf 0 -

1

l +1-topology2 to the corresponding subset of some 11:-solution.^3

We first prove, in dimension 3, the following strengthening of Theorem 28.8.

THEOREM 28.22 (Existence of 3-dimensional singularity models at all curvature

scales). Let (M^3 ,g(t)), t E [O, T), be a finite-time singular solution to the Ricci

flow on a closed 3-manifold. Given any sequence of points (xi, ti) E M x [O, T) with
Ri ~ R (xi, ti) -+ oo, define

gi (t) ~Rig (ti+ R"i^1 t).

Then there exists a subsequence of (M, gi (t), (xi,O)) which converges to some 11:-

solution (N!, h 00 (t)), t E ( -oo, OJ.

PROOF. By Theorem 28.6, there exists 11: > 0 such that (M^3 ,g(t)), t E [O,T),

is 11:-noncollapsed below the scale 1. Theorem 28.21 then says that for any j E N
and for such a 11:, there exists rj E (0, lJ such that if ij is chosen large enough so that
Rij 2:: rj^2 (this is possible since Ri -+ oo as i-+ oo), then the solution gij (t) on

(^2) Here I c:- (^11) denotes the least integer greater than or equ a l to c:- (^1).
(^3) 0f course, Bg(O) (xo, c (^112) ) = B 9 (to) (xo, (c:Q)-^112 ).