320 36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES
point of the developing neckpinch singularity and defining the arclength parameter
s relative to this choice, we are led to rescale the time and space coordinates in the
following way:
(36.15) T(t) := - log(T - t) ,(36.16) <J ( x, t ) := S ~ ( X l t) = e T /2 s ( x, t ).
vT-t
Further, since maximum principle arguments show that
1/;(x, t) 2: J(n - l)(T - t)
everywhere in the parabolic region, we are led to rescale the evolving geometric
quantity 1/;(x , t) as follows:( 36 ) U(<JT)·= 1/;(s,t)
'17
'. J2(n - l)(T - t)
One readily calculates the evolution equation for U(<J, T) to be(36.18) (<J ) (8uU)2
()TU= OuuU - 2 + nl 8uU + (n - 1) U + 1 ( 2 U - U 1) ,where I := J;^8 y/ d<J.
Motivated by Perelman's arguments that neckpinches in three dimensions be-
come cylindrical and presuming that this behavior extends to higher dimensions,
we make the first of the auxiliary (regional) ansatze:ANSATZ 36.3 (Parabolic Ansatz 1). In the parabolic region, as the singularity
time Tis approached (i.e., as t ---+ T, or , equivalently, as T ---+ oo), the quantity
U(<J, T) approaches unity uniformly.
Note that we may use this ansatz to define the extent of the parabolic region,
delineating it as that region in which the quantity V(<J, T) := U(<J, T) - 1 is suf-
ficiently small. In essence, then, the content of Parabolic Ansatz 1 is that there
indeed is a region in which this is the case. Within this region, it is useful to rewrite
(36.18) as an evolution equation for V(<J, T ) ; segregating terms which are linear and
nonlinear in V , we have(36.19) aTV =.CV+ N(V),
where
(36.20)(J.CV:= OuuV- 28uV + v
is the indicated linear operator and N(V) includes the higher terms. As a conse-
quence of Parabolic Ansatz 1, in that region the term N(V) may be neglected.
The operator £ is a familiar one from studies of neckpinches in mean curvature
fl.ow [15] as well as from studies of singularity formation in solutions of equations
of the form OtW = OxxW + uP [121]. The operator is self-adjoint in the space
2 .,2
L (~, e-T d<J) and has the pure point spectrum
k
(36.21) >..k = 1 - 2with the associated eigenfunctions being the Hermite polynomials hk(iJ). It follows
from standard Bernoulli (separation ansatz) considerations that solutions of the