412 8. APPLICATIONS OF THE REDUCED DISTANCE
there exists a neighborhood B 900 (e) ( q, ~ inj 900 (e) (q)) and a smooth function
F on K such that for each (J E (A -l, A) the function ·
q (x, e) ~ F (x, e) - fi ( <I>i o exp 900 (1,q) (x), (J)is. convex on B 900 (l,q) ( 0, l 6 inj 900 (l) (q)) C (TqM 00 , g 00 (1, q)). Since fi (·, ())
is differentiable almost everywhere,
\7 900 (1,q)'>i (x, e) --+ \7 900 (l,q) ( F (x, ()) - foo ( exp 900 (l,q) (x) , (J))a.e. on B 900 (1,q) ( 0, l 6 inj 900 (l) (q)) by Theorem D6.2.7 in [202]. Hence
\7 9oo(l,q)fi ( q>i 0 exp9oo (l,q) (x) 'e) converges to \7 9oo(l,q)foo (expgoo(l,q) (x) 'e)
a.e. on B 900 (1,q) ( 0, l 6 inj 900 (l) (q)) and \7 9 ri(e)fi ( i o exp 900 (l,q) (x), (J) con-
verges to \l 900 (e)foo (exp 900 (l,q) (x),e) a.e. on B 900 (1,q) (o, l 6 inj 900 (1) (q)).
Because q is an arbitrarily point on M 00 , we have proved that
(8.49) IVgr/il;ri(B) (<I>i (·) 'e)--+ J\7goofool~oo(B) (·, ())a.e. on M 00 for each (J E (A-^1 , A).
From 2~~ +J\7 Yri(e)fiJ^2 -R 9 ri (&)+~ = 0 and (8.49) we know ~~ (i (·), ())
converges a.e. on M 00 for each (J E (A-^1 , A). Since fi (i (-), ()) converges
to £ 00 (·, e) uniformly, we conclude
f)fi ( ( ) ). Ofoo ( )
()() <.I? i. ' (J --+ ()(). ' (Ja.e. on M 00 for each (J E (A-^1 ,A).
(ii) For any smooth compactly supported function cp(q, ()) 2:: 0 on M 00 x
( A-1, A), for i large enough we can extend cp (<I> i^1 ( q1) , (J) by 0 to a smoothfunction, still denoted by cp (i^1 (q 1 ), e), which has compact support on
M x (A-1, A). Using· the Lipschitz test function e-f.i(qi,B)cp ( i^1 (q 1 ), e) in
(7.146), we getand
{A { (f)fi · n)
j A-l j M f)(J + \7 9r/i · \7 9ri 'P - RgTi + 2(JX e-f.i(i:f!i(q),B)<p (q, ()) dμ<p;gri(B) (q) d(J 2:: 0.Taking the limit i--+ oo, we obtain (8.48). The lemma is proved. D