1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. MATCHED ASYMPTOTIC STUDIES OF DEGENERATE NECKPINCHES 323

As determined above, the evolving geometry at the tip is modeled by solu-

tions of (36.30). To relate these solutions to the Bryant soliton function B(r), one

imposes

ANSATZ 36.6 (Tip Ansatz). In the tip region, as t approaches the singularity

time, the scaling function f(T) satisfies the estimates f(T) »et and8rr = o(f^3 ),

and the derivative of the geometric function Z satisfies the estimate BtZ = o(f^2 ).

Taken together, these assumptions imply that the left-hand side of equation (36.30)

may be neglected, so Z may be presumed to be a solution of F,,[Z] = 0.

2.6. The outer region.

The simplest model for a degenerate neckpinch is reflection symmetric across

its equator and has just one neck and two bumps. For the matched asymptotic

analysis of such models, one has sequentially (from the right pole to the left pole) a

tip region, an intermediate region, a parabolic region, an intermediate region, and

a tip region. However, more generally, and for less symmetric neckpinch models,

while one side of the parabolic region includes an intermediate region and a tip

region, on the other side is an outer region, which we now describe. Generally, the

outer region does not lead to another neckpinch.

If the arc length coordinate s is chosen with its zero at the center of the parabolic

region, the outer region is identified as the region where the rescaled coordinate

(]" = et s approaches -oo. The geometry in this region is controlled by the quantity

'lfJ(s, t), and without imposing any further ansiitze, one determines by matching

with quantities in the adjacent parabolic region that

(36.33) 'ljJ(s, t)^2 ~ 2(n - 1) [(T-t) - (~)

2

],

where k is the integer (2: 3) introduced in the Parabolic Ansatz 2 and where c is

the constant appearing in the expression (36.27) for W.

It follows from this expression for 'ljJ in this region that if k is even, then the full

diameter of the solution, from pole to pole, decays to zero at the rate Isl :::; c(T-t)i.

If rather k is odd, then the distance from the equator to the right pole (at which

the singularity is forming) decays at this same rate.

2.7. Matching at regional overlaps.
As discussed above, the regional analyses and ansiitze lead to regional approx-
imate solutions with free parameters: In the parabolic region, expression (36.22)
for V ( (]", T) has the free parameter bk (along with the choice of the integer k); in
the intermediate region, expression (36.27) for W(p, t) involves the free function
of integration c(t); and in the tip region, expression (36.32) for Z(I') has the free
scaling factor a and also depends implicitly on the free expansion factor function
f(T) which relates --y(s, t) and 'lfJ(s, t). The regional solution in the outer region, as
described here, has no free parameters.
These free parameters are crucial for matching the regional approximate solu-
tions across the regional overlaps. Only by restricting their relative values does one
obtain a globally consistent approximate solution. As shown in [11], one finds the
following: (i) the integer k ;::: 3 may be freely chosen; (ii) matching the parabolic
and intermediate regions requires that c(t) be constant and that bk =-~ck; (iii)

matching the tip and the intermediate regions requires that a = k(~~l) and that