1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

42 28. SPECIAL ANCIENT SOLUTIONS

estimate for Xo = 'Y (2kc-^1 l^2 ) , where 0::::: k::::: l zc:1;2 - ~J ~ f'..^1 That is,

e

Volgoo(t) Bgoo(t) ( 0 , r) ;:::: L Volgoo(t) Bgoo(t) ( 'Y ( 2kc-^1!^2 ) ) c-^1 /^2 )

k=O

;:::: (R + 1) /),c-n/^2

;:::: ( 2c=1/2 - ~) K,c-n/2. D

Partially motivated by Corollary 2 8. 13 , we have the following. Stronger moti-

vation comes from work of P erelman, which implies that the conjecture is true in

dimension n = 3; see Theorem 28. 20 b elow.

OPTIMISTIC CONJECTURE 28.14 (R and Vol of noncompact singularity mod-

els). Any noncompact singularity model defined on a time interval (-oo,w) must

have scalar curvature bounded from above on each interval (-oo, wo], where wo < w.

Consequently, th e volume growth of any noncompact singularity model must be at

least linear.

To b etter characterize when a singularity forms, it is n atural to consider the

following.

OPTIMISTIC CONJECTURE 28.15. For any finite-time singular solution g (t),

t E [O, T), on a closed manifold Mn , its scalar curvature must satisfy

sup R = oo.

Mx(O,T)

By the Hamilton-Ivey estimate (see Theorem 9.4 in Volume One) , this is

true when dim M = 3. Zhou Zhang [454] h as proved this conjecture for solutions

to the K a hler- Ricci fl.ow on closed manifolds.

We also have the following criterion for a singular solution to have unbounded

scalar curvature.

LEMMA 28.16 (Criterion for long-time existence under bounded scalar curva-

ture). Given a finite-time singular solution (Mn, g (t)), t E [O, T), on a closed man-

ifold, if there are no Ricci fiat singularity models associated to it, then sup M x [O,T) R

= 00.

PROOF. By Theorem 28.8, there exists an associated singularity model

(M~, 9 oo (t)), which is nonfl.at by definition. Suppose tha t supMx[O,T) R < oo.

Then , by the standard lower estimate for R , we have supMx[O,T) IRI < oo. This im-

plies that the scalar curvature of g 00 (t) is R 900 (t) = 0. By aRa7<'l = .6. 900 (t)Rg 00 (t)+

iRc 900 (t)J

2

, this implies Rc 900 (t) = 0, contradicting the hypothes is. D

A primary goal in studying singularity models is to understand their geometry
as well as their topology. For example, one anticipa t es the need to understand the
geometry at infinity of noncompact singularity models. The following m ay b e a
small step in this direction.
PROBLEM 28.17. What can one say about the asymptotic cones of noncompact
singularity models?

(^1) Her e, L · J : IR-+ Z denotes t h e floor function; i .e., LxJ is the greatest integer less than or
equa l to x. In particular, LxJ + 1 > x.