48 2. SPECIAL AND LIMIT SOLUTIONS
To prove the claim, fix some to such that 'ljJ (-,to) is a Morse function, and
let its smallest critical value be attained at xo. Then the Implicit Function
Theorem implies that there exists a smooth function x ( ·) defined for t near
to such that x (to) = xo and 'l/;x (x (t), t) = 0. Taking the total derivative,
one obtains
:t 'I/; ( i ( t) , t) = 'l/;t ( X ( t) , t) + 'l/;x ( X ( t) , t) !~ ( t)
= 'l/;t(x (t), t)
n-l
='l/; 55 (x(t),t)- ()'
rmin t
The first inequality in the claim follows from this when one recalls that
'l/; 55 2: 0 at a neck. To prove the other inequality, recall that R = 2nKo +
n (n - 1) Ki, where Ko= -'l/; 55 /'lj; and Ki= (1 - 'I/;;) /'l/;^2. So at a neck,
[n - 1 R] n - 1 R
'l/;ss = -'l/;Ko = 'ljJ -2-Ki - 2n = --::;;;;-- 'ljJ 2n.Since the inequality R > 0 is preserved by the fl.ow, the second inequality
follows. 0
The final steps in this part of the proof show that no singularity occurs
on the polar cap: the region between the last bump and the pole. To do this,
we first use the tensor maximum principle to show that the Ricci curvature is
positive there. We shall invoke the following slight modification of Theorem
4.6.
LEMMA 2.33. Let (Nt,fJNt, g (t) : 0 :St < T) be a smooth I-parameter
family of compact Riemannian manifolds with boundary. Let S and U be
symmetric (2, 0)-tensor fields on Nt such that S evolves by
a
at S = 6.S + U * S,
where U S denotes the symmetrized product (U S)ij = Uikskj + SfUkj·
Suppose that infpENt S (p, 0) > 0 and that S (q, t) > 0 for all points q E aNt
and times t E [O, T). If (U * S) (V, ·) 2: 0 whenever S (V, ·) = 0, then
inf S (p, t) 2: 0.
pENt, tE[O,T)
Once one observes that the hypotheses imply that S can first attain a
zero eigenvalue only at an interior point of Nt, the proof is almost identical
to what we shall present in Chapter 4. We will use the proposition to prove
that 'ljJ is strictly concave on the polar caps.
LEMMA 2.34. If 'l/; 55 (x, 0) :S 0 for x*(O) < x < 1, then 'l/; 55 (x , t) :S 0 forx*(t) < x < 1 and all 0 < t < T.
PROOF. First we show that the Ricci tensor satisfies
a
(2.56) at Re= 6. Re +U *Re,