1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

40 28. SPECIAL ANCIENT SOLUTIONS

Regarding t he sectional curvatures of a ncient solutions, B .-L. Chen h as proved

the following.

THEOREM 28.5 (Ancient 3-dimensional solutions h ave sect 2'. 0). Any complet e

3-dimensional ancient solution must have nonnegative sectional curvature.

We omit the proof. We just rema rk that it may b e of interes t to find sharper

forms of the local estimate that B.-L. Chen uses to prove this theorem.

2. Properties of singularity models

In this section we recall how the study of singular solutions on closed manifolds

leads us to consider both i;;-solutions and GRS. We show that singularity models

with bounded scalar curvature h ave at least linear volume growth. In dimension

3, we prove (1) the existence of singularity models at all curvature scales , (2) t hat

singularity models must have bounded curvature, (3) that the asymptotic cone of

a i;;-solution is either a h alf-line or a line.

2.1. Singularity models are noncollapsed below all scales.

We now turn to singularity models. Using his entropy monotonicity formula,

in [312] Perelma n proved that all finite-time solutions on closed ma nifolds h ave the

following fundamental property (see also Theorem 6.74 in Part I).

THEOREM 28 .6 (No local collapsing using the scalar curvature). Let (Mn, g (t)),

0 :::; t < T < oo, be a solution to the Ricci flow on a closed manifold and let

p E (O,oo). There exists i;; = i;;(n, g(O) ,T , p) > 0 such that ifxo EM, to E [O, T),

and r 0 E (0, p) are such that the scalar curvature R satisfies the condition R '."::: r 02

in the open geodesic ball Bg(to) (xo, ro), then

Volg(to)Bg(to) (xo, r) > i;; JOr r. a ll O < r < ro.

rn - -

By definition, we say that g (t), t E [O, T), is i;;-noncollapsed below the scale p.

A finite-time solution (Mn,g(t)), t E [O, T), to the Ricci flow on a closed

manifold is called singular if sup M x [O,T) I Rm I = oo, where Rm denotes the Rie-

mann curvature t ensor. We are most interested in such solutions. By the work

of Sesum [365], a ny finite-time singular solution on a closed m anifold must have

SUPMx[O,T) JRcl = 00.

To understand singular solutions, we rescale and take limits.

DEFINITION 28 .7 (Finite-time singularity model). Given a finite-time singular

solution g ( t) on a closed manifold Mn, an associated singularity model is a

complete nonflat ancient solution to the Ricci flow which is the pointed Cheeger-

Gromov limit ofrescalings gi (t) ~ K ig (ti + Ki-^1 t), where ti--+ T and the rescaling

factors satisfy K i --+ oo.

P erelman's no local coll apsing theorem and Hamilton's Cheeger- Gromov-type

compactness theo rem for solutions to Ricci flow imply the following (a slightly

different statement is given in Theorem 19 .4 in Part III).

THEOREM 28.8 (Existence of singularity models with bounded curvatures). Let

(Mn, g (t)), fort E [O, T), with T < oo, be a singular solution to the Ricci flow on

a closed manifold. Then there exists (xi, t i ) with t i --+ T and K i ~ IRml (xi, t i) --+