152 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS
differentiable int, ft~ (t) ;::: 0 and ~ (t) :S: 1. Fix any compact subset K C Moo
and t E [a 00 , w 00 ). Then, by Fatou's lemma,
/ (47r (w 00 - t))-n/^2 e-eoo(x,t)dμ 900 (t) (x)}[(
= / liminf ((47r (wi - t))-n/^2 e- l,(x,t)) dμ9,(t) (x)
j K i--+oo:S: liminf / (47r (wi - t))-n/^2 e-l;(x,t)dμ9,(t) (x)
i--+oo } K
:S: lim inf ~ ( t)
i--+oo
::::: 1.Hence V 00 ( t) :S: 1 is well defined for t E [ a 00 , w 00 ). (Note that this holds as long as
the limit f, 00 exists, in particular, under the more general Type A assumption on
gi( t).)
(2) Monotonicity of V 00 (t). By the definition of derivative,(30.74) dVoo (t) = lim V(^00) (t + h) - V (^00) (t)
dt h--+0 h
= (47r)-~ lim J (q, t , h)dμ 900 (t) (q),
h--+0 Moo
provided the limit exists and where
~ ~ ( e- eoo(q,t+h) dμgoo(t+h) (q) - cfoo(q,t) )
(q,t,h)~ n () n ·
h (w 00 -t-h)2 dμg 00 (t) q (w 00 -t)2
Applying the standard formula ftdμ = -Rdμ to this, we see that at any point
(q, t) where 8f 00 /8t exists (in particular, for each t, a.e. on M 00 ), we have
e-eoo(q,t) ( n f)f, )
lim<I>(q,t,h)= u ( )- !:J00
-R 900 •
h--+0 (w 00 - t) 2 2 W 00 - t ut
Claim 1. We have(30.75) ~~ J <I>(q, t, h)dμg 00 (t) (q) = J ~~ <I>(q, t , h)dμg 00 (t) (q) l
Moo Moo
so that the evolution of the reduced volume based at the singular time is given by
(30. 76)
dVoo _uf e-eoo(q,t) ( n 8f 00 )
dt(t) = (47r) 2 Moo (woo - t)~ 2 (woo - t) - Bt -Rgoo(t) dμgoo(t)(q).PROOF OF CLAIM 1. Clearly (30.76) follows from (30.75). Now, as in the proof
of Theorem 8.20 in Part I , equation (30.75) sh all follow from Lebesgue's dominated
convergence theorem and estimating (q, t, h) by an integrable function on M 00 ,
independent of h sufficiently small. We proceed to do this.
By Proposition 30.18, we have that e- e^00 has quadratic exponential decay and
that I !l..fff-1 has quadratic growth. This, together with the Bishop- Gromov volume
comparison theorem, implies that (q, t , h) is bounded by an integrable function
on M 00 , independent of h sufficiently small. This proves Claim 1 (see pp. 396 - 398
in Part I for further details).