1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

174 4. PROOF OF THE COMPACTNESS THEOREM

4.4. Construction of the limit. We are now ready to construct the

limit manifold (M~, g 00 ). Topologically, we take the direct limit

M~ ~ ~B (ok,2k),

where the directed system comes from the maps Wk· Note that since wk are
approximate isometries, they must be open embeddings. Hence Moo is a
Hausdorff space by Lemma 4.17.
We recall the embeddings Ik : B (Ok, 2k) -+ M 00 defined in (4.5). The
coordinate maps Hf: : Ea -+ B (Oki 2k) induce coordinate maps H~,k ~
Ik o Hf: : Ea-+ M 00 • Note that the transition maps

(4.13)

are C^00 diffeomorphisms (when the domain and range are suitably restricted)

and hence they induce a C^00 structure on M 00 • Furthermore, since Wk,r are

isometries between gk,oo and gk+r,oo, we easily see that the transition maps

are isometries and there exists a metric g 00 on M 00 such that

Jkgoo = gk,oo·

We now show that {(Mk, gk, Ok)} converges to (M~, goo, Ooo). Given

a compact set KC M 00 , it must be contained in h [B (Ok,2k)J for some

k > 0 and hence must also be contained in I.e. [ B ( O.e., 2.e.) J for all .e z k. We

now claim that for every p there exists ko = ko (K, p) such that for any c > 0

sup Iva (goo - (!;;1 )* gk) I < c

xEK 9=

for all a ~ p, k z ko. This follows by pulling this expression back by Ik to

get

Iv~= (goo - (Ii:

1

) * gk) 19= = Ilk [ v~= (goo - (Ii:

1

) * gk)] II/;9=

= lv~k,= (gk,oo - gk)I 9k,oo

and by using Proposition 4.40. Hence the maps If;^1 satisfy the requirements

of Definition 3.5 and we have shown the following.

PROPOSITION 4.41 (Convergence to a limit). {(Mk,gk,Ok)} converges

to (M~, goo, Ooo)·

Furthermore, we can show that the limit metric is complete.

PROPOSITION 4.42 (The limit is complete). The metric g 00 is complete.

PROOF. Any closed geodesic ball B C M 00 is contained in the image

h [ B (Ok, 2k) J for some k E N. Recall that h is an open embedding, which