1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

170 4. PROOF OF THE COMPACTNESS THEOREM

Define the local versions of Fk£;r by

Gf P.;r ~ ( fif)-

1

o FkP.;r o Hf.

We may pull back to jjjf3 via the map fif to get

L q/k o Hf (X) exp;~ (X) Fk£,(3 (X) = 0,

a::;A(r) · k£;r

where now the map exp is with respect to the metric gf = ( fif) * gg.

To see the limiting behavior of Fk£;r when k, .e are large, we note that
since Proposition 4.34 implies for each f3 that we have

Fk£,f3 ___,. id(J when k, .e ___,. oo,

then by Proposition 4.54, we have G~fr ___,. id(J on any compact subset of Ef3
in C^00 • We have proved the following.'

PROPOSITION 4.35 (The maps Fk£;r converge to id in a sense). For every
r > 0, c > 0, and p E NU {O} there exists ko = ko (r, E,p) such that for
f3:SA(r),

I \JP ( G~P.;r - id(J) I ::; c

for all k, .e 2:: ko, where \7 and I· I are the covariant derivative and norm with

respect to the Euclidean metric on Ef3.

As a corollary we have the following.

COROLLARY 4.36 (FkP.;r is a local diffeomorphism). There exists ko =

ko ( r) such that if k, .e 2:: ko, then Fk£;r I B.a is a diff eomorphism for each

k
(3:SA(r).

PROOF. If G~fr is sufficiently close to the identity map, then it must be

injective since its derivative is nonsingular. D

Now we turn to proving that given (c,p) and r, fork, .e large enough, Fk£,r
is an (c,p)-pre-approximate isometry. First we have the following general
result.

LEMMA 4.37 (Limit of almost-identity pullbacks). Let ¢k : U ___,.UC Rn

be diffeomorphisms, let id: U ___,. U be the identity map, and let {hkhEN and

h 00 be Riemannian metrics on U. Suppose hk and h 00 are uniformly equiv-

alent to the Euclidean metric for all k E N and their derivatives (covariant

derivatives with respect to the Euclidean metric) are uniformly bounded. If

¢k ---t id and hk ---t h 00 in C^00 uniformly on compact sets, then for every
c > 0, p E N, and compact set K c U, there exists ko = ko (c,p, K) such

that if k 2:: ko, then

sup sup l\lr (¢khk - hoo)I (x) :Sc,

o::;r::;pxEK