- THE LIMIT MANIFOLD (M~,9 00 ) 165
Finally, we want to ensure that the Bk are embedded geodesic balls and
to ensure that the requirements of Proposition 4.53 are satisfied, so we need
that
205e^2 0cO A. a -< 410e^2 0cO A. [ra] k < -! 3 inJ' (xa) k 'which is ensured if
1230e^2 0cO >.. [rk] :S μ [rk, io]
or by (4.6), equivalently,1230e^20 cO /D:::; 1, i.e., 20c'C:::; Dlog (D/1230),
where c' ~ a·min {io, l}n (so that by (4.8) we have c ~ c' /D). Since DlogD
goes to infinity as D --t oo, we can choose D large enough to satisfy this
inequality and also such that c = c' / D is less than 1. Finally, we can make D
large enough so that the balls have radius less than 7r / ( 6.jCQ) , and hence
are convex by Corollary 4.47, and so that the balls have radius less than
c1/ .JCo as in Proposition 4.32.
To complete the proof of statement (5) of Lemma 4.18, we need to showthat the number of (3 such that Bf n Bk =I= 0 is fewer than I (n, Co). For
any (3 such that Bf n B'k =I= 0, Bf c Bk. As in the proof Lemma 4.21,
we can estimate the volume of B'k from above by a multiple of (A.at and
the volume of each B~ from below by a multiple of (A.at; this will give the
bound I (n, Co).
4. The limit manifold (M~, g 00 )
We can now construct limit overlap maps of balls and limit metrics; we
shall use these to take a direct limit and find the limit Riemannian manifold
(M~,g 00 ). Let Bk, Bk, Bk, A (r), K (r), and I (n, Co) be as in Lemma
4.18.
4.1. Local metrics on balls, transition functions, and their lim-its. Given r > 0, consider the ball B (Ok, r). For this section we shall always
assume that k 2:: K (r). Note that by Lemma 4.18, B (Ok,r) is covered by
the finite coliection of balls B'f:, where a :::; A (r). For each a :::; A (r) we
shall construct maps
Ffc: Bk --t Me
using the exponential maps of Mk and Me. In addition, we wish to average
these maps in such a way that the maps limit to the identity, in a sense, as
k,£ --t 00.
We first look at the metrics in normal coordinates in balls and obtain
convergence of the metrics locally. Choose linear isometries
Dt, : ]Rn --t Tx'kMk·We can then define all of the diffeomorphisms
Hk a.. Ea --t Ba k'