440 9. BASIC TOPOLOGY OF 3-MANIFOLDSsectional curvature. By the strong maximum principle, the universal covers
of the ancient solutions either split as the product of a surface solution with
IR or have positive sectional curvature in which case they are diffeomorphic
to either 53 or JR^3.^2 In the case of splitting, Hamilton proved that the an-
cient surface solution is either a round shrinking 52 (and the universal cover
of the singularity model is hence geometrically a shrinking round product
cylinder; by definition we say that in this case a neck singularity forms) or
it has a backward limit which is the cigar soliton. Note that Perelman's no
local collapsing theorem rules out the last case of a cigar. In the case when
the universal cover of the singularity model has positive sectional curvature
and is diffeomorphic to JR^3 , the covering is trivial. This ancient solution is
either Type I or has backward limit which is a steady Ricci soliton on a
topological ffi.^3.
In the latter case, the asymptotic scalar curvature ratio is infinite, and
by dimension reduction, there exists a sequence of points tending to spatial
infinity for which the corresponding dilations of the solution limit to an
ancient product solution, which again must be a shrinking round cylinder.^3
In this case the singularity model is expected to be the positively curved
and rotationally symmetric Bryant soliton and the forming singularity is
expected to be a degenerate neckpinch. In summary, we should have that
at the largest curvature scale the dilations yield the Bryant soliton, whereas
at lower scales dilations yield round product cylinders. This agrees with the
fact that the dimension reduction of the Bryant soliton, and more generally
a 3-dimensional gradient steady soliton with positive curvature which is 11,-
noncollapsed at all scales, is a round product cylinder.
On the other hand, the former case of a Type I ancient solution with
positive sectional curvature, if it exists, also dimension reduces to a round
cylinder. Thus a consequence of Hamilton's 3-dimensional singularity for-
mation theory and Perelman's no local collapsing theorem is that if a finite
time singularity forms on a closed 3-manifold, then either M is diffeomor-
phic to a spherical space form or a neckpinch forms.
In [186] Hamilton studies 3-dimensional singularity formation by con-
sidering regions in the solution where the scalar curvature is comparable to
its spatial maximum. He studies the regions where the scalar curvature is
not comparable to its spatial maximum by the technique of dimension re-
duction. For example, when a 3-dimensional steady Ricci soliton singularity
model forms, Hamilton proved an injectivity radius estimate to obtain a sec-
ond limit which is either a shrinking round product cylinder or the product
of a cigar with R (Again no local collapsing rules out the latter case.) The
(^2) When the universal cover of the singularity model has positive sectional curvature
and is diffeomorphic to S^3 , M admits a metric with positive sectional curvature (e.g., g (t)
fort large enough) and hence is topologically diffeomorphic to a spherical space form. In
this case the singularity model must be geometrically a shrinking spherical space form
with its underlying manifold diffeomorphic to M.
(^3) With the help of no local collapsing.