- ALMOST /<;-SOLUTIONS i35
STEP 2. We shall prove the following.
Claim 2. For To (w) defined by (20.20) below, we haveti '.S-To(w).
When ti ::::; -1, Claim 2 is true since To ( w) < 1.
When ti > -1, we will prove Claim 2 by contradiction. If Claim 2 is not
true, i.e., if ltil ::::; To (w), then there exists Yo E B 9 (o) (xo, ~) and t2 E [ti, OJ
such that
(20.18)and
9 19
40
:S dg(t) (yo,xo) :S
80fort E [ti, t 2 ].
Applying Proposition 20.6(a) tog (t), t E [ti, OJ, we get
(20.19)
for y E Bg(t) (xo, i) and t E [ti, 0].
Now we apply Theorem 18.7(2) tog (t), t E [ti, t2], with the Ricci curva-ture upper bound oni^4 Bg(t) (xo, fo)UBg(t) (yo, fo) c Bg(t) (xo, i) to estimate
dg(t) (yo, xo), where fo ::::; lo is to be chosen below. We get
~ dg(t)(Yo, xo) ~ -2 (n -1) (-
32
(c (5-nw) + B (
5
-nw)) fo +! ).
ut t - ti roSince -ti ::::; 1, we may choose fo = (t-~/
12
::::; lo to obtain8at dg(t) (yo, xo)
~ -2 (n - 1) ( C (~;~w) (t - ti)i/^2 + ( B (~;~w) + 80) (t - ti)-i/^2 ).
Integrating this inequality in time from ti to t2, we have
d 9 (tz) (yo, xo) - d 9 (t 1 ) (yo, xo)~ -2 (n - 1) ( C (~;~w) ltil^312 + ( B (~~nw) + 160) ltili/^2 ) ,
where we used t2 - ti :S lti I·
Define(20.20)
(^14) The Ricci curvature bound follows from (20.19).