334 7. THE REDUCED DISTANCEan exhaustion {Uk}kEN of M 00 by open sets with Poo E Uk and diffeomor-
phisms <I>k : Uk ~ Vk ~ <I>k (Uk) c Mk with <I>k (Poo) = Pk such that
(uk,<I>/:: (gk(T)lvk)) ~ (Moo,9oo(T)) in C^00 on compact sets in Moo x
[O,T].LEMMA 7.66 (£under Cheeger-Gromov convergence). Under the setup
above, for any (q, r) E M 00 x (0, T), we have
.eg(oo Poo, 0) ( q, r) = k-+oo lim .eg(pk k, 0) ( <I>k ( q) 'f').The convergence is uniform on compact subsets of M 00 x (0, T). Further-
more, the convergence is uniform in C^00 on compact subsets of the open set
of points at which £(; 00 ,o) is C^00 •PROOF. Let';:/: [O, f'] ~ M 00 be a minimal £-geodesic, with respect to
g 00 , joining p 00 to q. By the convergence of the sequence of solutions {gk ( T)}
in C^2 on compact sets, we have
£(; 00 ,0) (q, f') = 2 ~£goo (';;/) = 2 ~ kl!_.1! Cgk ( <]?k 0 ';;/)
2: limsup£g(pk o) (k (q) ,f').
k-+oo ki
Next we prove the opposite inequality. Since for any compact set JC, C
Moo,
(7.102)we know for every q E M 00 that
dgk(r) (pk,k (q)) = d<P;':,(gklvk)(r) (Poo,q) ~ dg 00 (r) (Poo,q)
as k ~ oo. For sufficiently large k, let 'Yk : [O, f'] ~ Mk be the minimizing
£-geodesic with respect to 9k from Pk to k (q). By Lemma 7.13(iii) and (i),
we have
£bk) = L(;k,o) ( k (q), f') S C2,
and for any T E [O, f'] ,
dgk(O) (Pk,"fk (T)) s C2
for k large enough, where C2 is a constant independent of k and T (depend-
ing on n, f', T, Co, dg 00 (r) (Poo, q)). Hence
compact set K1 C Moo independent of k. For k sufficiently large, we have
L(;k,o) (k (q) ,f') = C<Pj:,(gklvk) (/;^1 ('Yk)). By (7.102), we get
lim inf .eg(; o) ( k ( q), f') =
1
k-+oo k, 2y £ T lim k-+oo inf£,,,_* "'k ( gk I vk ) ( <l>/;^1 o 'Yk)
1
= 2v Elim T k-+oo inf Cg 00 ( <l>/;^1 o "fk) 2: £g(p^00 oo, O) ( q, f').