164 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONSThen by (30.109) we haveso that \79^00 (tlm 00 (t) is a parallel gradient vector field.
Claim.(30.116) v~ (t) = v;; (t).
Proof of claim. (1) m 00 (t) is constant. Then \79^00 (tJet, (t) = \79^00 (t)g~ (t) and
6. 900 (t)et, (t) = 6. 900 ctJR~ (t). In this case, (30.111) implies et, (t) = e~ (t), which
in turn yields (30.116).
(2) m 00 (t) is not constant. Then \79^00 (tlm 00 (t) is nonzero and by the de Rhamholonomy splitting theorem we have the following. The shrinker (M 00 ,g 00 (t)) is
isometric to (Nn-^1 ,h(t)) x (IR,ds^2 ) , where (N, h(t)) is isometric to {x E M 00 :
m 00 (x, t) = O} with the metric induced by g 00 (t). Moreover, there exist a, b E IR
depending on t with a -=/=- 0 such that(30.117) m 00 (y, s , t) =as+ b
for y EN and s ER From (30.109) we can now show that there exists ¢(t) : N-+ IR
such that
s2 Ae~ (y, s , t) = ¢ (y, t) - - +as+ b
4t(s )2 A
= ¢ (y, t) -
2
t - at + b + a^2 t^2
for some a, b E IR and
By translating the coordinate on IR and absorbing the constant b + a^2 t^2 into ¢ , we
may assume that a = b = O; i.e.,
s2e~ (y, s, t) = ¢ (y, t) -
4t.
Hence, by (30.112) and (30.117), we haveThis implies
o = Iv e;t, ( t) 12 + et,t ( t) - Ive~ ( t) 12 - e~/ t)
I a) 2 as+ b
= 2a \ ve~ (t)' OS +a + -t-b
= a2 + -.
t