- REDUCED DISTANCE OF TYPE A SOLUTIONS 139
whose graph joins (;(q), l) to (pi,wi).^2 By Lemma 7.32 in Part I , we have that
the gradient of Li ( ·, l) with respect to gi(t) is given by
(30.19)
In order to estimate(30.20)
it suffices to estimate the speed of /'ii we now proceed to do this.
STEP 2. An ODI for IVLilg, (l'i (t) 't). Let xi (t) ~ 1'~ (t) and let
1
(30.21) Yi (t) ~ VWi - tXi = -2 \7 Li (!'i (t), t).Our goal is to estimate IYi(l)I. Using the £-geodesic equation for the solution 9i (t)v~ xi - -
21
\7 R 9 - 2 Reg (Xi) - 2 (1
- i Wi - t) xi = o
(see (7.32) in Part I, with T = wi -t), we compute that
(30.22)
Hence
(30.23)where we used (30.8) and (30.4).
STEP 3. Estimating IV Lil~, at some time. To apply the oor (30.23) to estimate
ll'H l) I, we just need to estimate 11'~ ( t) I for some t E [t, wi]. We haveL (q, l) = £ 9 , (!'i) = 1w; ( VWi - t R 9 , (!'i (t), t) + ~ IY; (t)l:,(t)) dt
and (using (30.8) and r < ~)
- Jr r' ,,/wi -t R r ' Cn
9 , (!'i(t), t) dt "5. Jr ,,/wi -t (wi _ t)7' dt
< --2Cn ( w -a ).:l.-r^2.
- 3 - 2r^00 00
Hence, by the pointwise bound (30.9) for Li,
(30.24) 1 Wi ~ IYi (t)l:,(t) dt
=Li (q, t) -1Wi VWi -tR9i (!'i (t) 't) dt
"5,.D ,(^2) Note that l'i([t,wi]) need not be contained in 4(Ui)·