2. REDUCED VOLUME AT THE SINGULAR TIME FOR TYPE I SOLUTIONS 153
We wish to use (30.76) to prove that d~? (t) ;:::: 0.
Claim 2. The RHS of (30.76) is nonnegative.
PROOF OF CLAIM 2. By (30.33) we h ave
(30.77)i
t
2
( [ (^2 fJ£^00 n ) J
ti }Moo (\lf.oo, \l<p)g= + \j\lf.oolg 00 - Dt -Rg 00 + 2 (woo_ t) <pJdμg 00 dt 2 0for any nonnegative C^2 function <p on M 00 x [t 1 , t 2 ] with compact support. Since
we h ave (30.70), (30.71), and (30.72) and in particular e-£^00 is Lipschitz and decays
quadratic exponentially, inequality (30.77) holds by a pproximation for <p = e - £=.
Since
we conclude that
This proves (2).
(3) The equality case. Suppose that V 00 (t 1 ) = V 00 (t2) for some ti < h Then
from (30.78) and (30.30), we conclude that
. Of.oo 2 n
(30.79) Qoo =:= -Dt -6'.g 00 foo + j\lf.00 1900 - Rg 00 + 2 (woo_ t) = 0
in the sense of distributions on M 00 x [t 1 , t2] · This implies that
(30.80)
where D::C ~ -gt -6. 900 + R 9 =, in the sense of distributions. Applying standard
parabolic regula rity theory to (30.80), we conclude that f. 00 is C^00 on M 00 x [t 1 , t 2].
Thus (30.80) holds in the classical sense.
On the other hand, recall from (7 .94) in Part I that we h ave
-2oii + lviil
2- R -(t) + ~ = 0.
at §;(t) g, Wi - t
Hence, by R em ark 30.12, we obtain
(30.81)
in the sense of distributions. By (30.79) and (30.81), we have
(30.82) V 00 ~ ((w 00 -t) (Rg= +26'.g=f.oo -j'Vf.ool~ 00 ) +f.oo -n) (w~::t/2
e-e=
= (w 00 - t) (Foo - 2Qoo) /2
(w 00 - tt
= 0.