1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

186 4. PROOF OF THE COMPACTNESS THEOREM

Furthermore F ( x) is smooth and its derivatives \le F ( x) are bounded by

constants depending on the bounds for [\lei μi ( x) [ with fi ::::; £, the bounds

for I \lei Fi ( x) I with fi ::::; £, and n, £, and Co, ... , Cc+i ·

PROOF. Fix xo E W. By assumption, there exists a small neighborhood

Vx 0 CW of xo, io, ro, and ji, ... ,jm such that Fjk (x) EB (Fio (xo), ro) for

k = 1, ... , m and all x E Vx 0 and such that μj (x) = 0 for all j =/::. ji, ... ,jm

and x E Vxo· By Lemma 4.50, for x EV we can define Fx 0 (x) on Vx 0 to be

cm( μJI. (x) , ... ,μJm. ) {Fj 1 (x), ... , Fjm (x)}

which is the composition of cm(μ· 21'""'' μ· Jm, ) {qJ1' ... , qjm} and μjk = μjk (x),

qjk = Fjk (x). From the uniqueness of the center of mass it is easy to see

that for any two points x1, x2 E W, Fx 1 (x) = Fx 2 (x) on Vx 1 n Vx 2 • Hence

this defines F : W ---+ M.

We now prove the second part of the lemma. Let xo E W; then, on Vx 0 ,

F (x) is the composition of cm(μh, ... ,μjm) {%1' ... , qjm} and μjk = μJk (x),

%k = Fjk (x). It follows from the chain rule and Proposition 4.51(i) that

F ( x) is smooth on Vxo and that F ( x) on Vx 0 has the required derivative

bounds. D

We also used the following convergence property.

PROPOSITION 4.54 (Average by cm of maps limiting to id limits to id).

Let B1 C B2 be two open subsets of "JRn and let gk, where k EN, be a family

of Riemannian metrics on B2. Assume all derivatives of the curvatures of

gk are uniformly bounded and gk ---+ g 00 on any compact set in B2 in C^00 •

Suppose Fk : B1---+ B2, for a= 1, ... , A, are sequences of smooth maps such

that Ff: ---+ id uniformly on compact sets in C^1 for each a as k ---+ oo. Let

μk, be partitions of unities for each k on B1. For any compact set K C B1

we can define Fk : K---+ B2 fork sufficiently large by letting Fk (x) be the

center of mass of Ff: (x) with weight μk, (x) with respect to metric gb i.e.,

Fk ( x) is defined by

A

L μk, (x) exp_F~(x) Ff: (x) ~ 0,

a=l

where the exponential map expFk(x) is with respect to gk. Then Fk converges

to id as k ---+ oo uniformly on any compact set in B 1 in C^1.

PROOF. Because some of lv~v~ cm(μi, ... ,μk) {q1, ... , qk}I, Jal+ l,61 2: 2,

do not approach 0 as qi, ... , qk ---+ q*, we will not prove this proposition
using the composition employed in the proof of Proposition 4.53. By Remark
4.52(ii), cm^9 (k μk 1 ( ) x , ... ,μk A( x )) { FJ (x), ... , Ff (x)} can be made arbitrarily close
to cm^9 (μk^001 ( x ) , ... ,μk A( x )) { FJ (x), ... , Ff (x)} on any compact set x EK C B1 in
C^00 when we choose k large enough. On the other hand, fix xo E B1 and let
w be normal coordinates centered at xo in (B 2 , g 00 ). It follows from Remark