- THE LIMIT MANIFOLD (M~,g 00 ) 167
To obtain the local convergence of Jff3 as k -+ oo, we take some fur-
ther subsequences. Since the Jf^13 are Riemannian isometries between gk ~(.Hf)* 9k and §f for each k and since the derivatives of the metrics are
bounded, we have that for each pair a, f3 ::; A ( r) such that Bk n Bf =f. 0
for all k 2: K ( r) , there exists a subsequence such that the Jff3 converge to
a limit transition map
J~: Ea-+ E/3
in C^00 uniformly on compact sets (by Corollary 4.11). In fact, J~ is a
Riemannian isometry between g~ and ~. Since Jf 13 is a restriction of
Jf^13 , we also have that Jf^13 converges to a map J~. We then diagonalizethe sequences so that Jf^13 converges for every a and (3. Notice that since
Jf a o Jf^13 = id13 : Ef3 -+ Ef3, the identity embedding, we must have that
Jg: 0 J~ = id13.
4.2. Constructing approximate isometries Fkf!.;r of large balls in
Mk into Me. We can now construct approximate isometries Fkf!.;r between
the ball B (Ob r) c Mk and an open set in Me for sufficiently large k and
£. (We shall use some results about the center of mass given in Section 5 of
this chapter.) The following is the main result of this subsection.
PROPOSITION 4.33 (Existence of an approximate isometry on a large
ball). For every r > 0, c: > 0, and p > 0 there exists ko = ko (r,c:,p) > 0
such that fork, R > ko there is a diffeomorphism
Fkf!.;r : B (Ok, r)-+ Fkf!.;r (B (Ok, r)) c Me
which is an (c:,p)-approximate isometry.Let Ffe: Bk-+ Bf (from a ball in Mk to a ball in Me) be defined by
Fkl!. a 7. Ha I!. 0 (Ha)-1 k.
Roughly speaking, we construct the desired map Fkf!.;r by averaging the local
maps Ffe for a ::; A (r). Notice that in terms of the (inverse) coordinate
systems ( Ef3, He) and ( Ef3, iif), where f3 satisfies Bf n Bk =f. 0, the map
Ffe corresponds to the mapFkf!.,/3 a ·. E/3 -+ E_, f3between Euclidean balls defined by
Ffe,/3 ~ ( fif)-
1
o Ffe o He= (iif)-
1
o Hf o (.Hf)-^1 o Hf(4.10)