1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

150 4. PROOF OF THE COMPACTNESS THEOREM

then mapping by the exponential map to Me. We do this for each ball. This
is done in subsection 4.1 below.
STEP C: Use nonlinear averaging to glue together the maps Ffj_ to obtain
maps Fke : B (Ok, 2k) ~Me which take Ok to Oe (done in subsection 4.2
below). By taking subsequences, we can ensure that compositions of Fk,k+l
are approximate isometries which are getting closer to isometries as k goes
to oo.
STEP D: We form the limit manifold M~ as the direct limit of the
directed system { Fk,k+l : B (Ok, 2k) ~ B ( Ok+l, 2k+l)}. The coordinates of
B (Ok, 2k) then form coordinates for the limit M 00 , i.e., for each coordinate
Hf: : gx ~ B (Oki 2k) there is a coordinate for the limit manifold defined
as
H~ k = Ik 0 Hf: : gx ~Moo,
'

where h is the inclusion of B (Ok, 2k) into M 00 • Furthermore, for each

coordinate Ea of B (Ok, 2k) there are Riemannian metrics 9'k,e which are
obtained by the pullbacks Pt.e9e· Since Fke are approximate isometries, the
sequence is equicontinuous and thus by the Arzela-Ascoli theorem we can
find a convergent subsequence and get local metrics 9~ k· It is not hard
to see that these metrics form a Riemannian metric 900 ' on M 00 via the
coordinate charts H~ k· We can then show that the limit metric is complete

and (M 00 , 900 , 000 ) s;tisfies the theorem, where 000 is the equivalence class

of the base points in the direct limit. This is done in subsection 4.4 below.
At many points in this construction we will take a subsequence; to sim-
plify notation, at each stage the sequence will be re-indexed to continue to
be k.

2. Approximate isometries, compactness of maps, and direct limits

In this section we shall introduce some basic concepts which are essential

to the construction of the limit manifold (M~, 900 , 000 ).

2.1. Approximate isometries. In the following, the notation /T/ 9
means the length at a point of the tensor T with respect to the Riemannian
metric 9.

DEFINITION 4.1 (Approximate isometry). For any 0 < E: < 1 and p E

NU { 0} , a smooth map : (Mn, 9) ~ (Nn, h) is an ( c, p )-pre-approximate
isometry if

sup /<I>*h - 9/ 9 ~ c,

xEM

sup/ (~D-^1 )* 9 - h/ ~ c,

xEN h