- NECKLIKE POINTS IN FINITE-TIME SINGULARITIES 269
subsequence of points {yi} in M^3 , orthonormal frames {Fi} at (Yi, ti), and
an orthonormal frame F 00 at (y, 0) such that
. Rm (yi, ti)
ihm -+oo IR m ( Xi, ti ) I = Rm^00 (y, 0),
where Rm 00 ~ Rm [g 00 ] and the curvature operators are represented in the
respective orthonormal frames. Because Rm 00 (y, 0) > 0, we have Yi E Mf^0
for some co > 0 independent of i. Because ti ---) T, Lemma 9.13 implies that
0
Um [(T - ti) (R (yi, ti)+ p)]" IRm (yi, ti) 12 2 = 0.i-+oo (R(yi,ti)+p)
Then since (T - ti) (R (yi, ti)+ p) 2: c > 0 and R (Yi, ti) ---) oo, we conclude
that 0
IRmoo (y, 0) 12
(Roo (y, 0))^2.
0
Since y E M^3 was arbitrary and R 00 > 0, this implies that Rm 00 (., 0) = 0.
Then Schur's lemma implies that g 00 (0) is a metric of constant positive
sectional curvature on M~. By Myers' Theorem, M~ is compact, hence
diffeomorphic to M^3. It follows that g (t) has uniformly positive sectional
curvature for t close enough to T, hence that g (t) limits to a metric of
constant positive sectional curvature. DREMARK 9.14. Mao-Pei Tsui has suggested an alternate proof of the fact
that F remains bounded if for every c > 0, there are T < T and 6 > 0 such
that no c-essential 6-necklike points exist after time T. The argument uses
the enhanced maximum principle for systems (Theorem 4.10) and resembles
our first proof of Theorem 6.30. It is simpler than but basically equivalent
to the argument given above in Lemma 9.13.ALTERNATE PROOF THAT F IS BOUNDED. Recall that >. 2: μ 2: l/ and
that>.+μ+ v + p = R + p 2: c > 0. Since
d
dt (A + μ + v + P) = >. 2 + μ2 + v2 + >.μ + Av + μv,we have
d μ2 + 1/2 + μv - PA-d log ( >. + μ + v + p) = A + >..
t +μ+v+p
Fix a point where A > v and define
A - l/
<I>~ (T- t)" (A+μ+v+p)l-c'Then since ft log (A - v) = >. + v - μ,we compute
d E pA - (μ^2 + v^2 + μv)
- log = EA - - - - (μ - v) + (1 - E).
ill T - t A+μ+v+p