- NECKLIKE POINTS IN ANCIENT SOLUTIONS 273
(When 'Y = 1, K corresponds to the bound in the definition of ancient Type
I singularity models.) Since the scalar curvature of (M^3 ,g(t)) is positive,
the function 0
G = jtj'Yc/2 1Rmj2
· R2- c
is well-defined. Since the sectional curvatures are positive, we have the
estimates
G :S CRc jtj'Yc/ 2 :S CKE jt j--yc/ 2 '
which show that G is bounded for all times -oo < t ::; 0 and satisfies
(9.10) lim max G (x, t) = 0.
t-+-OOxEM3Note that our choice of G is similar to taking T = 0 and p = 0 in the function
F studied in Theorem 9.9, except that we now consider the possibly smaller
power (T - t)'Y = jtj'Y. This modification does not significantly harm the
evolution of G: we will once again be able to show that if there are no
essential necklike points, then there exists E > 0 small enough that G is a
subsolution of the heat equation, hence that its maximum cannot increase.
The advantage of the modification is that we can then use ( 9 .10) to conclude
that G = 0, hence that (M^3 , g(t)) is complete and locally isometric to a
round 53 , hence that it is compact and globally isometric to a spherical
space form 53 /r.
If there are no ancient c-essential J-necklike points on the time interval
(-oo, T], then for every x E M^3 and t E (-oo, T) either
(9.11) IRm (x, t)I · ltl < c
or we have
(9.12) IRm-R(B@B)I > 8jRml
for every unit 2-form e at (x, t). Taking <p = (-t)'Yc/^2 jR
0mj^2 , 'ljJ = R, ex= 1,
and /3 = 2 - E in Lemma 6.33 and estimating as in Lemma 9.11, we get the
differential inequalitygt G :S ~G +^2 (l; c) (VG, \1R)+2J,
where
jtj'Yc/2
J=. --[ clRm^0 l^2 ( IRml^2 - -'YR) -P ]. R3- c 4 lt l
and P ~ 0 is defined in (6.39). Fix any (x , t) with t < T :S 0. If (9.11) holds
there with c ::; 'Y /8, then we have
IRml^2 - 4[tT 'YR ::; R ( IRml - 4 'Y lt l ) < R ( ftT c - 4 'Y ltl )
and hence "(E