1547671870-The_Ricci_Flow__Chow

5. THE NECKPINCH 61

Since Lemma 2.31 implies that 7./Js 2 0 when 0 ::; s ::; s(t), one may then
estimate at any x E [xD(t), x(t)] that

(2.66)

rt r(x) (7.f; )2

s*(t) 2 s(x,t) 2 n lo lo ; dsdt 2 p(t).

In particular, p( t) is bounded above by the distance from the equator to the

bump.

We are now ready to prove that the singularity occurs only on the equa-

torial hypersurface.

PROPOSITION 2.45. If the diameter of the solution g(t) remains bounded

as t / T, then 7.f;(s, T) > 0 for all 0 < s < D/2.

To establish this result, let to E (0, T) and 6 > 0 be given. For E > 0 to

be chosen below, define

l'.(s, t) = c{s - [p(t) - p(to - 6)]}.

By (2.66), the finite-diameter assumption implies that p(T) < oo, hence

that

sup [p(t) - p(to - 6)]

to-8<t<T

becomes arbitrarily small when to - 6 is sufficiently close to T. Proposition

2.45 is thus an immediate consequence of

LEMMA 2.46. If p(T) < oo, then for any to E (0, T) and 6 E (0, to),

there exists E > 0 such that v = 7./Js satisfies

v(s, t) 2 1'.(s, t)

for all points 0 < s < D /2 and times to ::; t < T.

PROOF. Since 1'.(0, to) < 0 and v(s, to) > 0 for all s E (0, s*(to)), one

may choose E1 such that if 0 < E < E1, then v(s, to) > 1'.(s, to) whenever

0 ::; s ::; D /2. So if the lemma is false, there will be a first time f E (to, T)

and a point s E (0, D /2) such that v(s, f) = l'.(8, f). At (s, f), one then has

(2.67) Vt ::; l'.t = E (as at -p ')

as well as v = l'., V 8 = l'.s = E, and V 88 2 l'.ss = 0. Hence

(n-2) (n-1)( 2 ) (n-2) (n-1)(1- 1'.^2 )

Vt= V 88 + 7./J VV 8 + 7.j; 2 1 - V V 2 E 7./J V + 7./J 2 V.

Whenever 0 < E < c 2 = ./2/D, one has l'.::; Es< 1/./2 for all 0::; s::; D/2

and t E [to, T). Then because v 2 0 for s E (0, D/2) C [O, s* (t)], one