- NECKLIKE POINTS IN FINITE-TIME SINGULARITIES 263
If for every c > 0, there exist some T < T and fJ > 0 such
that there are no Type I c-essential 6-necklike points after
time T, then the normalized solution g ( t) converges to a
spherical space farm.Our strategy will be to modify the second proof of Theorem 6.30 in order
to show that the pinching of the curvature operator improves sufficiently
under the fl.ow when this hypothesis holds. Recall that we considered the
function
(9.4)
0. 1Rml^2
f =:= R2-c: ,
and saw that f stays bounded for c sufficiently small when (M^3 , g) has
strictly positive Ricci curvature. The first difficulty we face is that we no
longer know R > 0. But if we choose constants p 2: 0, c > 0 so that
R + p 2: c > 0 at t = 0, we have R + p 2: c for 0::; t < T by the maximum
principle, because
8ot (R + p) = t:,. (R + p) + 2 IRcl^2.
Thus we may replace R by R+p in the denominator of (9.4). This introduces
an additional bad term in the evolution equation for (9.4). To remedy this,
we introduce the time factor (T - t)c: and define
0 0
(9.5) F =:= (T - t)c: IRml2 = [(T - t) (R + p)]c: IRml2.
(R + p)2-c: (R + p)2
Since we must allow for arbitrary initial metrics, we shall have to work
harder to show that F stays bounded. However, the multiplicative factor
(T - tr helps, because its time derivative is negative. In fact, a key step inproving the theorem will be to show that F .. 0 as t .. T if (M^3 , g) has no
0
essential necklike points. Notice that (R + p)-^2 1Rml^2 is scale invariant if
we ignore the constant p, and that (T - t) (R + p) is invariant under Type
I rescaling.Our first step is to derive an analog of Lemma 6.34.LEMMA 9.10. There exists a constant C = C (g (0)) such that
IRml ::; C (R + p).PROOF. We may assume without loss of generality that v ::; μ::; >.; and
since IRml scales like R, we may assume that minxEM3 v (x, 0) 2: -1.
Set A = e^6 and B = A/c, so that B (R + p) 2: A. Let N denote the
number of negative sectional curvatures at a point (x, t). There are two
cases: If - A ::; v, it is easy to see that