- THE NECKPINCH 55
To complete the proof we must deal with possibility that Lmin (0) < e^2.
In this case, we consider a rescaled solution
g (t) = >,g (±)
of the Ricci fl.ow. Denoting its sectional curvatures by k 0 - k and
K1 = L, we have Lmin = >,-^1 Lmin and k = >,-^1 K. Thus the choice
A = e-^2 Lmin (0) > 0 implies that L (0) 2': e^2 , whence the preceding ar-
guments apply to the metric g. We conclude therefore that max{n - 1, F}
is nonincreasing, where F = ( k / L) log L. Translated back to the original
metric g ( t), this implies thatmax { n - 1, ~ log ( Lm~: ( 0 ) L) }
does not increase with time, as claimed. DThe bound for F we obtained above is an example of what is called a
'pinching estimate' for the sectional curvatures, and is a foreshadowing of
similar results we will see in Chapters 6 and 9.
Our next result will let us compare the actual radius 'l/J (x, t) near a
neck with the radius J2 (n - 1) (T - t) of the cylinder soliton which is its
singularity model.LEMMA 2.40. Let g (t) be as in Proposition 2.36. Fort E [O, T), choose
xo (t) E W (t) so that
'l/J (xo (t), t) = rmin (t).
DefineO" = S ( X, t) - S (XO ( t) , t) ,
Then there are constants 8 > 0 and C < oo such that for t sufficiently close
to T, one has(2.61) l< •'+' ' <l+-----
(^1) • (x t) C ( O" )^2
- r min ( t) - - log r min ( t) r min ( t)
(2.62) ---'l/J (x, t) ~ C O"^1 og-------;::========= O"
r min ( t) r min ( t) J -log r min ( t) r min ( t) J -log r min ( t)for 2rmin (t) J-log rmin (t) ~ O" ~ rmin (t)l-o.
PROOF. Let r=; denote a small positive number to be chosen below. We
regard t as fixed, and consider the neighborhood of the neck xo(t) in which'l/J ~ r=; and l'l/Jsl < r=;. In this region, one always has L 2': (1 - r=;^2 )/r=;^2 , so that
there exists a constant C < oo such that
12 - logLmin (O)I ~CL.