- CLASSICAL SINGULARITY THEORY 475
THEOREM A.69 (3d Type I - existence of necks). If (M^3 ,g (t)) is
a Type I singular solution of the Ricci flow on a closed 3-manifold on amaximal time interval 0 :::; t < T < oo, then there exists a sequence of
points and times (xi, ti) with ti ~ T such that the corresponding sequence
of dilated solutions (M^3 , .9i (t), xi) converges to the geometric quotient of a
round shrinking product cylinder S^2 x R
For the rest of this section we consider Type Ila singular solutions. In
this case we invoke Perelman's no local collapsing theorem (see Chapter 6).
This has the following two effects on Hamilton's theory. It enables one to
apply the compactness theorem to the dilation of Type Ila singular solu-
tions. It rules out the formation of the cigar soliton as a product factor in
a singularity model.
Classical point picking plus the no local collapsing theorem yield the
following result (see Proposition 8.17 of [111]).
PROPOSITION A.70 (Type Ila singularity models are eternal). Choose
any sequence Ti /' T. For a Type Ila singular solution on a closed manifold
satisfying(A.29) IRml:::; CR+ C
and for any sequence {(xi, ti)}, where ti~ T, satisfying^6
(A.30) (Ti - ti) R (xi, ti) = max (Ti - t) R (x, t) ,
Mx[O,T.]
the sequence (M, §i (t), Xi), where
(A.31) §i (t) ~Rig (ti+ Hi^1 t) with Ri ~ R (xi, ti),preconverges to a complete eternal solution (.M~, § 00 (t), x 00 ), t E (-oo, oo ),
with bounded curvature. The singularity model (.M 00 , § 00 (t)) is nonfiat, 11,-
noncollapsed on all scales for some 11, > 0, and satisfies
sup R (§ 00 (t)) = 1 = R (§ 00 ) (x 00 , 0).
Meo x ( -oo,oo)If n = 3, then the singularity model has nonnegative sectional curvature.
In particular, we have that if ( M^3 , g ( t)) is a Type Ila singular solu-
tion of the Ricci fl.ow on a closed 3-manifold, then by Theorem A.64, the
singularity model (.M~, § 00 (t)) obtained in Proposition A. 70 is a steadygradient soliton. Since (.M 00 , § 00 (t)) is nonfiat and 11,-noncollapsed on all
scales for some 11, > 0, the sectional curvatures of § 00 (t) are positive.^7 Now
by Theorem A.58, the asymptotic scalar curvature ratio of (.M 00 , § 00 (t)) is
(^6) This is a special case of the point picking method described in subsection 4.2 of
Chapter 8 in Volume One.
(^7) In the splitting case, we obtain a cigar, contradicting the no local collapsing theorem.