476 A. BASIC RICCI FLOW THEORY
equal to infinity. Thus we can apply dimension reduction, Theorem A.59,
to get a second limit which splits as the product of a surface solution and a
line. This second limit is a shrinking round product cylinder 52 x R^8
Hence, as a consequence of Hamilton's singularity theory and Perelman's
no local collapsing theorem, we have the following result, which complements
Theorem A.69.
THEOREM A.71 (3d Type Ila - existence of necks). If (M^3 ,g (t)) is
a Type Ila singular solution of the Ricci flow on a closed 3-manifold, then
there exists a sequence of points and times (xi, ti) such that the corresponding
sequence (M, gi (t), xi) converges to a round shrinking product cylinder 52 x
R
A precursor to the above result is Theorem 9.9 in Volume One, which
basically says that even for a Type Ila singular solution on M^3 x [O, T),
there exists a sequence of points (xi, ti) with ti ---+ T whose curvatures sat-
isfy (T - ti) JRm (xi, ti) J 2: c for some c > 0 independent of i and at the
points (xi, ti) the curvature operators approach that of 52 xlR. after rescaling.
However Theorem 9.9 in Volume One does not directly imply the existence
of a cylinder limit because, for the sequence (M, 9i (t), Xi), it not a priori
clear that the curvatures are bounded in space at finite distances from Xi
independent of i, even at time 0. The reason for this is that globally, the
curvature of 9i (0), whose norm is 1 at Xi, may be unbounded since the solu-
tion is Type Ila whereas the point (xi, ti) may, for example, have curvature
(T - ti) JRm (xi, ti)J ::SC for some C < oo.independent of i.
Since finite time singularities are either Type I or Type Ila, we obtain the
existence of necks for all finite time singular solutions on closed 3-manifolds.
(^8) 0therwise we again get a cigar limit.