1547671870-The_Ricci_Flow__Chow

40 2. SPECIAL AND LIMIT SOLUTIONS

anything we have presented thus far, but illustrate many important ideas
that will be developed further in the chapters to come.
The main results of [6] may be summarized as follows.

THEOREM 2.15. If n ?: 2, there exists an open subset of the family of

SO ( n + 1 )-invariant metrics on sn+^1 such that any solution of the Ricci fiow

starting at a metric in this set will develop a neckpinch at some time T < oo.

The singularity is Type-I (rapidly-forming). Any sequence of parabolic dila-

tions formed at the developing singularity converges to a shrinking cylinder

soliton (2.41) uniformly in any ball of radius o ( J-(T - t) log(T - t)) cen-

tered at the neck.

Any SO(n + 1)-invariant metric on 5n+l can be written as

(2.42) g = <p^2 d X^2 +^0 '// /,2 gcan

on the set ( -1, 1) x sn, which may be identified in the natural way with

the sphere sn+l with its north and south poles removed. The quantity

?j;(x, t) > 0 may thus be regarded as the 'radius' of the totally geodesic

hypersurface { x} x sn at time t. It is natural to write geometric quantities

related to g in terms of the distance

s ( x) = fox <p ( x) dx

from the equator. Then writing gs = ~ gx and ds = cp dx, one can write the

metric (2.42) in the nicer form of a warped product

(2.43) g = ds^2 + 1/;^2 gcan ·

Armed with this notation, one can make a precise statement about the

asymptotics of the developing singularity.

THEOREM 2.16. Lets (t) denote the location of the smallest neck. Then

there are constants 5 > 0 and C < oo such that for t sufficiently close to T

one has the estimate

(2.44)^1 + o(l) <. ?j;(x, t) <^1 + C (s - s)^2

- J2(n - l)(T - t) - -(T - t) log(T - t)

in the inner layer Is -sl :<:::; 2)-(T - t) log(T - t), and the estimate

1/;(x,t) C s-s

1

s-s

(

2

.4

5

) )T - t :<:::; J -(T - t) log(T - t) og J-(T....., t) log(T - t)

in the intermediate layer 2J-(T - t) log(T - t) :<:::; s - s ::=:; (T - t)(l-o)/^2.

The estimates in the theorem are exactly those one gets when one writes

the evolution equation (2.47) satisfied by ?/; with respect to the self-similar

space coordinate a= (s - s) /)T - t and time coordinate T = -log (T - t)

and derives formal matched asymptotics. For this reason, it is expected that

these estimates are sharp.