396 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY
From (G.8) we have By (yo,c:-^1 +c:) C N2ro (f (Bx (xo,c:-^1 ))); hence by
(G.9), f : Bx (xo, c:-^1 ) -t By (yo, c:-^1 + c:) is a 2c:-isometry.
The above discussion implies (using Lemma G.12)LEMMA G.15 (Pointed Gromov-Hausdorff distance and c:-isometries).Let (X, dx, xo) and (Y, dy, Yo) be two pointed metric spaces. If
d~H ((X,dx,xo), (Y,dy,yo)) < c:,
then there exists a 2c:-isometryf: (Bx (xo,c:-^1 ) ,dx) -t (By (Yo,c:-^1 +c:) ,dy)
with f (xo) =Yo and there exists an 2c:-isometryg: (By (yo,c:-^1 ) ,dy) -t (Bx (xo,c:-^1 +c:) ,dx)
with g (yo) = xo.
A sequence {(Xi, dxi, xi)}iEN is said to converge in the pointed
Gromov-Hausdorff distance to (Y, dy, yo) ifi--+oo _lim d~H ((Xi, dxi' Xi) '(Y, dy' Yo)) = o.
1.2.3. Gromov's precompactness theorem.DEFINITION G.16 (Precompact topological space). A subset Sin a topo-
logical space Xis (sequentially) precompact if, for every sequence of points
in S, there exists a subsequence which converges to a point in X.
The Gromov precompactness theorem says the following (see The-
orem 5.3 of [78]). Given K E ~ and n ;:::: 2, let 9J1;tK be the collection of
pointed complete Riemannian manifolds (Mn,g,O) ~ith Re;:::: (n -l)K.
THEOREM G.17 (Gromov precompactness theorem, I). The subcollection
9J1;tK' in the collection of pointed metric spaces, is precompact with respect
to the pointed Gromov-Hausdorff distance.
One of the ideas of the proof of the Gromov precompactness theorem
is to use the Bishop-Gromov volume comparison theorem. Another is the
following general sufficient condition for compactness (see Theorem 7.4.15
of [18]).
THEOREM G.18 (General criterion for precompactness). Suppose that a
collection 9J1 of compact metric spaces is uniformly totally bounded, that
is,
(1) there exists D < oo such that for every X E 9J1 we have the diam-
eter bound diam X ::;_ D,(2) for every c: > 0 there exists N (c:) EN such that for every X E 9J1
there exists an c:-net Sc X consisting of at most N (c:) points.