324 36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHESf(T) = e(l-f-)^7 • With these restrictions imposed, two free parameters are left: the
choice of the integer k 2 3 and the choice of the constant c. For any choice of
these parameters, one obtains a globally consistent approximate Ricci fl.ow solution
which models a degenerate neckpinch.
As constructed, these approximate solutions become singular at time T at the
right pole. One readily calculates their rate of curvature blow-up:
c
(36.34) sup IRm(x, t)I rv (T )2-2/k.
xESn+l - t
For any choice of k 2 3, the indicated blow-up rate corresponds to a Type II
singularity. It is interesting to speculate whether this quantization of blow-up rates
is a real feature of degenerate neckpinch singularities of Ricci fl.ow or if it is ratherjust an artifact of the particular approximate models constructed in [11]. We note
that in the study [439] of the asymptotics of degenerate neckpinch singularities on
noncompact manifolds, this quantization of blow-up rates does not occur.
3. Ricci flow solutions with degenerate neckpinch singularities
It is known from [127] that Ricci flow solutions which develop degenerate neck-
pinch singularities and have Type II curvature blow-up behavior exist. The numeri-
cal work and formal matched asymptotics work discussed above indicate what some
of the detailed features of such solutions could be. These studies do not, however,
guarantee that Ricci fl.ow solutions with these features exist. In this section we
describe work which shows that indeed they do exist. In particular, we show that
for every one of the approximate formal solutions modeling degenerate neckpinch
singularity formation (as discussed in §2 above), there exist rotationally symmet-
ric Ricci flow solutions which asymptotically approach this formal solution and
consequently share its asymptotic properties.
The formation of a degenerate neckpinch singularity is not expected to be a
stable property of Ricci flow solutions, so one does not expect to be able to use
sub and super solution methods to prove that Ricci fl.ow solutions approach each
of the (n, k, c)-parametrized approximate formal solutions. However, as shown in
[12], the closely related Wazewski retraction method [433] does the job. We note
its use to prove a similar result for mean curvature fl.ow solutions [15].
The idea of the Wazewski retraction method, as applied to the present problem,
is the following: We consider the space M(Sn) of rotationally symmetric metrics^5
on sn and represent the Ricci flow solutions of such metrics as trajectories in
M ( sn) x JR^1. A particular choice of one of the approximate solutions constructed
above in §2 via formal matched asymptotics is also represented by a trajectory in
M ( sn) x JR^1. Reflecting the parametrization of these approximate solutions, we
label them as .9{n,k,c}(t).
For each choice of an approximate solution .9{n,k,c} (t) and for any positive E,
we construct a tubular neighborhood 3{n,k,c;€} of that solution in M(Sn) x JR^1
which has the following features: (i) as t-+ T (the singularity time for .9{n,k,c}(t)),
3{n,k,c;€} progressively narrows so that any trajectory contained within it approaches
.9{n,k,c}(t) arbitrarily closely; (ii) the boundary of 3{n,k ,c;€} consists of three pieces
(^36 · 35) u.:::.{nJ'.:>~ ,k ,c;€} = u.:::.{nj'.l~-,k ,c;€} U J'.:>~0 u.:::.{n,k ,c;€} U u.:::.{nJ'.:>~+ ,k ,c;€} '
(^5) We note that in practice, each point in M(Sn) can be specified by a pair of (metric) functions
¢(s) and 'lj;(s).