1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. MATCHED ASYMPTOTI C STUDIES OF DEGENERATE NECKPINCHES 321

equ ation 8T V = .CV can be written in series form Lk bke>.kT hk (a-) for a sequen ce

of const ants bk. It is useful to recall that t h e le ading term in hk (a-) is o-k and that

for k even , hk(o-) involves only even powers of o-, while for k odd , it involves only

odd p owers.

If we were considering an initial value boundary value problem for the equation

8T V = .CV, we would use the boundary conditions and initial conditions to deter-

mine the constants bk. For the purposes here of modeling degenerate neckpinches,

we add a further ansatz.

ANSATZ 36.4 (Parabolic Ansatz 2). For some positive integer k :'.". 3, one

can write

(36.22) (o----+ oo),

with this term much sm aller t han one in the parabolic region.

We note that in [11], this ansatz is stated more phenomenologically, in terms

of certain aspects of the neckpinch singularity formation. We also note that since

we have defined the parabolic region as that in which V(o-, T) is small , it follows

from (36.22) that this region is delineated by the condit io.n io-1 « e(~-ilT.

2.4. The intermediate region.

As opposed to the parabolic region in which V is very small , the intermediate

region is marked by V being of order 1. Since it follows from Parabolic Ansatz

2 and the definitions of the Hermite polynomials that in the parabolic region the

leading order term in Vis comparable to e>-kT o-k, we m ay demarcate the neighboring

intermediat e region as that within which e>-kT o-k is of order 1.

To approximate the behavior of Ricci flow solutions in the intermediate region,

it is useful to work with the spatia l variable

(36.23)

(of order 1 in the intermediate region) and the metric variable

(36.24) W(p,t):=U(o-,T) (=l+V(o-, T)).

One may then write equation (36.18) in the form

(36.25) Ee k p W-~W- 2 2 ~w-^1 = e-T8tW- e(f-l)TF(W ' [) p w ' [) pp W)

where F is a functional of the indicated quantities.

Assume we now impose

ANSATZ 36.5 (Intermediate Ansatz). Ot W as well as F(W, Op W , Opp W) is

bounded in the intermediate region as T ---+ oo.

Then it follows that the right-hand side of equation (36.18) m ay be neglected, and
consequently the behavior of the flow in the intermediate region may be modeled
by solut ions of the ODE

(36.26)

p - 1 - 1 - -1

-8 k p W - -W 2 - - 2 W = 0.

One readily determines that the general solution to (36.26) is

(36.27) W(p, t) = J1 -(p/c)k,

with the constant of integration c(t) generally a function oft.