1547671870-The_Ricci_Flow__Chow

for c; > 0 small enough. D

5. Necklike points in ancient solutions

The objective of this section is to show that an ancient solution of posi-
tive sectional curvature is either isometric to a spherical space form or else
contains points at arbitrarily ancient times where its geometry is arbitrarily
close to the product of a surface and a line. We begin with some general
observations about ancient solutions - including the fact that every ancient
solution (of any dimension) has nonnegative scalar curvature.

LEMMA 9.15. Let (Mn, g (t)) be a complete ancient solution of the Ricci
flow. Assume that Rmin (t) ~ infxEMn R (x, t) is finite for all t :S: 0 and that
there is a continuous function K (t) such that !sect [g (t)]I :S: K (t). Then
g (t) has nonnegative scalar curvature for as long as it exists.

PROOF. Recall that the parabolic maximum principle works on noncom-

pact as well as on compact rpanifolds. Recall also that

~~ = b.R + 2 IRcl

2

2: b.R + ~R

2

.

So if R ever becomes nonnegative, it remains so for as long as the solution

exists. On the other hand, if Rmin (t1) < 0, then Rmin (to) < 0 for all to :S: ti,

and one has

n n

Rmin (t) > 1 > ----

• n (Rmin (to))- - 2 (t - to) - 2 (t - to)
  for all t > t 0. Since the solution is ancient, we may let to ""' - oo, obtaining

Rmin (t) 2: - lim ( n ) = 0.

to->- oo 2 t - to

D

REMARK 9.16. In dimension n = 3, we can strengthen this result, as we

proved in Corollary 9.8.

In Lemma 8.19, we derived a lower bound for the blow-up rate of so-

lutions that develop singularities in finite time. The analogous result for

ancient solutions is the following:

LEMMA 9.17. Let (Mn, g (t)) be an ancient solution with nonnegative

Ricci curvature. Then there exists a constant co > 0 depending only on n

such that

liminf ltl sup IRm (x, t)I 2: CO·

t->- oo xEMn