1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

182 4. PROOF OF THE COMPACTNESS THEOREM

5.2. Nonlinear averages. Let (Mn,g) be a complete Riemannian
manifold with sectional curvatures bounded above by K. Let p E M and

ql, ... , qk E B (p, r) , where r < min { i inj(p) , 6 )K}. Let μ1, ... , μk be

nonnegative numbers with μ1 + · · · + μk > 0. We define the center of mass
with weights μ1, ... , μk,

cm {q1, ... , qk} = cm(μ 1 , .. .,μk) {q1,. · ·, qk},

as the minimizer of

( 4.23)

¢: M ___,.IR,

1 k

¢ (q) ~ 2 Lμid2 (q, qi).

i=l

LEMMA 4.50 (Existence of center of mass). Let p EM and ql, ... , qk E

B (p, r) for some r < 6 JK. Suppose inj (q) > 3r for all q E B (p, r). Then

there exists a unique minimizer cm {q1, ... , qk} of¢ in M. Furthermore we

have cm {q1, ... , qk} EB (p, 2r) and

qi,.. lim .,qk-+q. Cm(μ^1 , .. .,μk) {q1, · · ·, qk} = q*

uniformly in μ1, ... , μk.

PROOF. It is clear that for any q EM \B (p, 2r), we have¢ (q) > ¢ (p).

Hence the minimizer of¢ exists and must be contained in B (p, 2r). Note

that if q EB (p, 2r), then q EB (qi, 3r). Since B (% 3r) CB( qi, 7r/ ( 2v1K)),

by Lemma 4.43, the functions q ~ !d^2 ( q, qi) are strictly convex in B (p, 2r).

Since the weights μi are nonnegative and μ1 + · ·+μk > 0, </>is strictly convex

in B (p, 2r). Hence the minimizer must be unique.

To see the last statement, we apply the first part of the statement

to B (q, r) in place of B (p, r), where r* is small. We get that when

ql, ... , qk E B (q*, r*), we have cm(μi,.. .,μk) {qi, ... , qk} E B (q*, 2r*) for all

(μ1, ... ,μk). D

By Lemma 4.43 we have

k

grad¢ ( q) = - L μi exp;-^1 qi.

i=l

The minimizer occurs at a point q where the gradient of ¢ is zero, so that

k

L μi exp;-^1 qi = 0.

i=l

The following proposition tells us about the derivatives of the center of mass.