158 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS
The following says that bounded scalar curvature, finite-time singular solutions,
if they exist, must be Type II.
THEOREM 30.23 (Type I singular solutions have unbounded scalar curvature).
Any finite-time Type I singular so lution on a closed manifold must have
sup R = oo.
Mx[O,T)
PROOF. By Theorem 30.22, there exists an associated singularity model
(M~,g 00 (t)) which is a non-Ricci fiat shrinking GRS. If SUPMx(O,T) R < oo, then
by t he proof of Lemma 28.16 (M 00 , g 00 (t)) must be Ricci fiat, a contradiction. D
In addition to I: 1 defined by (30.95), we have the following.
DEFINITION 30.24 (Sets of singular points). Given a finite-time singular solu-
tion (Mn, g (t)), t E [O,w), define the following sets of singular points:
I:R ~{PE M: liminf (w - t) R (p, t) > o},
t-tw
I: ~ {p E M : :3ti --+ w and Pi --+ p such that I Rm I (Pi, ti) --+ oo}.
From Definitions 30.21 and 30.24 it is immediately clear that I:R c I:1 c I:.
In other words, I: comprises the weakest notion of singular point, whereas I:R
comprises t he strongest notion of singular point.
The following surprising result, that for Type I solutions the above notions of
singular point are the same, is due to Enders, Muller, and Topping [104]. The
proof uses t he strong maximum principle and pseudolocality.
THEOREM 30.25 (Equality of sets of singular points). If (Mn, g (t)) is a Type
I singular solution to the Ricci flow on a closed manifold, then
(30.99)
PROOF. It suffices to show that I:1 C I:R and I: C I:1. Roughly speaking,
the issue is to obtain some local curvature control. As evidenced by the work of
Hamilton and, especially, Perelman, such control i s often obtainable indirectly by
limiting arguments.
(1) Let p EM - I:R. Then there exists Cj --+ 0 and tj E [w - Cj, w) such that
(30.100)
c-
Rg (p, tj) < -^1 -.
w - tj
Define Aj ~ (w - tj)-^1. By P roposition 30. 19 , corresponding to Pi= p and Aj, the
limit (M~,g 00 (t) ,p 00 ) in (30.85) is a complete shrinking GRS with bounded cur-
vature. By (30.100) we have R 900 (p 00 , -1) = 0. By applying the strong maximum
principle to the equation for
8
~~^00 , we obtain Rc 900 (t) = 0 for t ~ -1. Then, by
Chen and Zhu's uniqueness theorem (see [63]), we conclude that Rc 900 (t) = 0 for
t < 0. Since we are on a shrinker, this implies that Rm 900 (t) = 0. By Theorem
30.22, p EM - I:1.
(2) Let p EM - I: 1. Then
lim (w - t) IRml (p, t) = 0.
t-tw
Hence, for any Aj --+ oo, corresponding to Pi = p and Aj, the limit shrinker
(M~, goo (t) , p 00 ) in Proposition 30 .19 is isometric to Euclidean space. By the
