3. DEGENERATE NECKPINCH SINGULARITIES 325with no solution inside 2{n , , k c·<} , ever contacting 33+{ n , k ,c,E. } and with every solution
inside 3{n 1 kc·<} ' ' which contacts 83{-n, k ,C,E. } necessarily exiting immediately. As a
consequence of this property of 83{-n, k ,C,. } E , one readily determines that the exit
times (from 3{n ' k ' ' c·<}) for solutions that do not reach 83°{ n, k ,c,E. } depend continu-
ously on the choice of initial data for such solutions.
Presuming that a tubular neighborhood 2{n,k,c;<} with the properties just de-
scribed can be constructed, the Wazewski method argues as follows that a Ricci
fl.ow solution must asymptotically approach .9{n,k,c}(t): Fixing an initial time t 0 <
T (with to very close to T), one considers a k-dimensional subspace of the set
M(Sn) x {to} of all initial data sets for the Ricci fl.ow solutions of interest. One
then specifies a closed connected subset Bk of the parameter space for this set of
initial data sets, choosing it so that every initial data set 9[a] ( t 0 ) corresponding
to a parameter choice a E aBk lies in the tubular neighborhood instant exit set
83{n,k,c;<} and so that the map>.: a E aBk-+ 9[aJ(t 0 ) is not contractible. One also
chooses Bk so that the corresponding initial data sets 9[{3] ( t 0 ) for all /3 E Bk are far
enough away from the neutral set 83{n,k,c;<} so that the Ricci fl.ow solutions 9[{3) (t)
which evolve from this initial data have the property that so long as they remain in
3{n 1 k 1 c-<}, 1 they stay bounded away from 83°{n 1 k ,c,E. }. Consequently, these solutions
either remain in 3{n ' k ' ' c·<} or exit somewhere at 83{-n, k ,c,E. }.
Continuing the Wazewski argument, we now suppose that every solution 9[!3) (t)
(with /3 E Bk) leaves 3{n,k,c;<} at some time tf3. As noted above, exits can only hap-
pen at 83{-n , k ,c,E. } ) so it follows from this supposition that there is a continuous map
A-+ 83{n,k,c;<}. Moreover, since we know that initial data 9[f3J(to) with /3 E 8Bk
must be contained in 83{-n, k , c ,€. } and consequently have t13 = to, we see that the
map A must be an extension of the map >.. It then follows that >. must be con-
tractible. This violates the construction of >. as a noncontractible map. We hence
conclude that at least some of the Ricci fl.ow trajectories evolving from initial data
9[!3) (to) (with /3 E Bk) must remain in 3{n,k ,c;<} for all t E [to, T) and must therefore
asymptotically approach the formal approximate solution .9{n,k,c}(t). Consequently,
there are Ricci fl.ow solutions with the degenerate neckpinch singularity behavior of
these formal models.
It should be evident from this discussion that to implement the Wazewski re-
traction method in proving this degenerate neckpinch result, there are two primary
tasks that must be carried out. The most difficult of these tasks is verifying the
existence of the tubular neighborhoods 3{n,k,c;<} with their necessary properties. In
this survey, we do not discuss the details of the construction of these tubular neigh-
borhoods, nor the verification of their properties. We refer the reader to [12], where
this is done. We do note that in the portion of 3{n,k,c;<} corresponding to the tip
region, the intermediate region, and the outer region, sub and super solution bar-
riers can be constructed. These barriers prevent escape from that part of 3{n,k,c;<}
corresponding to those regions; hence the instant exit portion of the boundary of
the tubular neighborhood does not occur in these regions. In the parabolic region,
standard barriers do not exist, and exit from the corresponding portion of 3{n,k,c;<}
can occur. Consequently, the analysis of the portion of the tubular neighborhood
corresponding to the parabolic region is in fact where the most intricate estimates
are needed; we refer the reader to §6 of [12].