322 36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES
2.5. The tip region.
The numerical simulations discussed in § 1 of this chapter, as well as the results
in [127], indicate the Ricci flow for a critical initial geometry should asymptotically
approach the configuration of a Bryant soliton. It takes a bit of work to incorporate
this behavior into the approximate solutions of the formal matched asymptotic
analysis. One starts by noting that since the conditions for regularity at the right
pole require that 88 '1/;(s(l, t), t) = -1 for all t < T, it follows that 08 '1/; < 0 in a
neighborhood of that pole. It then follows from the implicit function theorem that
one may replace "s" by "'I/;" as a local spatial (radial) coordinate, and one may
write the metric in a neighborhood of the right pole in the form^4
(36.28)In this form, z is the function governing the configuration of the geometry of the tip
region. One readily verifies that the evolution of z is governed by the flow equation
(36.29) OtZ = F,p[z]
= ~ 2 { 'lj;^2 zz,p,p - ~('l/;z,p)^2 + (n - 1 - z)'l/;z,p + 2(n - 1)(1-z)z};
we note that the (nonlinear) differential operator F ,μ [ z J acting on z ('I/;, t) depends
explicitly on the coordinate 'I/;.
It is important to make one further coordinate change in the tip region: We
replace 'I/; by a new radial coordinate/, which is related to 'I/; by a yet to be deter-
mined time-dependent expansion factor f(T(t)); explicitly, 1(s, t) = r(T(t))'l/;(s, t).
We then set Z(!, t) := z('I/;, t) and calculate (from (36.29)) the evolution equation
for Z:
(36.30)here F'Y[·] is the operator defined in (36.29), but with 'I/; replaced by/, wherever it
appears.
As discussed in [11], the solutions of the ODE boundary value problem consisting
of the ODE F'Y[Z] = 0 together with the boundary conditions Z(O) = 1 and Z(oo) =
0 are closely related to the Bryant steady soliton metrics. More specifically, if one
writes the Bryant steady soliton metric [37] in the form
(36.31)then the solutions of F'Y [ Z] = 0 with these boundary conditions all take the form
(36.32) Z(!) = B (~)
for some positive scaling function a. We note that the function B(r) is not known
in explicit analytic form; it can, however, be represented arbitrarily accurately nu-
merically from the definition of the Bryant steady soliton, which essentially reduces
to the ODE boundary value problem under discussion here.
(^4) More precisely, as discussed in [11] one shows that there exists a negative definite function
y('lj;, t) such that &s(s, t) = y('lj;(s, t), t) and then sets z := y^2.