- PROPERTIES OF SINGULARITY MODELS 47
STEP 2. Existence of r::-necks. Fix a time t, which without loss of generalitywe may assume is equ al to 0, and fix a point 0 EM. Recall that the notion of an
r::-neck in a Riemannian manifold is given in Definition 18.26 of Part III.Claim 1. For any c > 0 there exists an r::-neck Nc: contained in (M^3 , g (0)).
PROOF OF CLAIM 1. Recall from Theorem 20.1 in Part III that the asymptoticscalar curvature ratio ASCR(g(t)) = oo for all t E (-oo, OJ. Hence, by Corollary
18.21 in Part III on dimension reduction for noncompact K-solutions with ASCR =
oo, there exists a sequence {xi} iEN in M such that the sequence 9i (t) = Rig (Ri^1 t),
where Ri ~ R 9 (xi, 0), on M x (-oo, OJ and based at the point (xi, 0), converges in
the C^00 pointed Cheeger- Gromov sense to the product of ffi. with a 2-dimensional
K-solution (which must be the shrinking round 2-sphere by a result of Hamilton (seeCorollary 9.19 in [77])). The existence of r::-necks in (M, g (t)), for any c > 0 and
any t E (-oo, O], now follows from the definition of Cheeger- Gromov convergence.
This finishes the proof of Claim 1.
Since Nc: is diffeomorphic to 52 x ffi. and since 8Nc: is smoothly embedded in
M ~ ffi.^3 ,^4 by the smooth Schonfl.ies theorem we h ave that M - Nc: has exactly
two components, a compact component Bc: diffeomorphic to a closed 3-ball and a
noncompact component Cc: diffeomorphic to 52 x [O, 1).
Furthermore, since Nc: is an c-neck, there exists an embedding (see p. 63 in
Part III)
(28.26) 'I/Jc: : 52 x [-c^1 + 4, r::-^1 - 4] ---+ N"and a radius rc: E (O, oo) such that r;^2 ¢;g(O) is r::-close in the clc:-'l+i_topology
to the standard unit cylinder metric gs2 + du^2. Without loss of generality, we may
assume that 'l/Jc:( 52 x {-c^1 +4}) is closer to 8Bc: and 'l/Jc:(5^2 x {c^1 -4}) is closer
to 8Cc:. Note that since g (0) has bounded curvature, the radius of Nc: satisfies
(28.27)where c > 0 is independent of c sufficiently small.^5
STEP 3. Rays and necks. First observe the following.Claim 2. For c > 0 sufficiently small, 0 E Bc:.
PROOF OF CLAIM 2. Since g (0) has positive sectional curvature (using thisonly at 0) and since each point of Nc: has a small sectional curvature,^6 for r:: > 0
sufficiently small we h ave that 0 ~ Nc:. Hence, if the claim is false, then there exists
a sequence E:i ':,i 0 such that 0 E Cc:; for all i E N. We may pass to a subsequence
{ ki} iEN SO that
(28.28) Nc:k, c C"kj for j < i
(in particular, the Ne:k, are pairwise disjoint). Indeed, suppose that we have chosen1 ~ k 1 < · · · < ki-l · Since /Ci ~ LJ (M - Ce:kj) is compact, the sectional curvatures
j<i4We may assume that N 0 ~ S^2 x IR, although this was not explicitly stated in Definition
18.26 of Part III.
5Note, however, that the radius r 0 is not uniquely defined for a given c:-neck.(^6) More precisely, the sectional curvatures satisfy
lim max (min { sectg(O) (Px ) : Px is a 2-plane at x}) = 0.
e-->0 xENe