- CENTER OF MASS AND NONLINEAR AVERAGES
This proves the first two convergences. Next we estimate
{)
\l qi oμi cm(μ1, .. .,μk) {qi, ... ' qk}.Since we have \l qi 8 ~i G = \l qi expq-i qi ----*id, by taking \l qi-derivative of
{) {)
~G+ \lqG · ~cm{qi, ... ,qk} = 0
uμj uμj
in ( 4.25) and taking the limit, we getid - (t μi) \J qj {){). Cm(μi, .. .,μk) {qi, ... , qk} = 0.
i=i μJis5This proves the third convergence. D
REMARK 4.52. (i) Note that in Euclidean space ~n, we have the follow-
ing formula for the center of mass
1
Cm(μ 1 , .. .,μk) {qi,···, qk}. = μi +... + μk (μiqi + ''' + μkqk) ·It is clear that there are many derivatives of the form
\l~\l~ cm(μ 1 , .. .,μk) {qi, ... , qk}, where ial + l,61 ?: 2,
whose lengths do not approach 0 as qi, ... , qk ---- q E ~n.
(ii) Suppose his another metric on M. From the proof of Proposition
4.51 and Remark 4.49(ii), it is not difficult to see that, as a function of
(μi, ... , μk, qi, ... , qk), the center of mass map cmg( μ1, .. .,μk ) {qi, ... , qk} is
close to cmh( μ1, .. .,μk ) {qi, ... , qk} on any compact set in C^00 when g is very
close to h on any compact set in C^00.
We can use the center of mass to average maps. We have the following.
PROPOSITION 4.53 (Averaging maps). Let (Nn, h) and (Mn, g) be Rie-
mannian manifolds such that all derivatives of the curvature of M are
bounded:
lv.eRml ~ C.e for.€.= O, 1, 2, ....
Let μi (x) , i = 1, · · · , k, be a finite sequence of smooth nonnegative functions
on N with compact support in Ui C N and with bounded derivatives. Let
W be an open set with closure W c LJiμii (0, oo). Let Fi : Ui ----* M be a
finite sequence of smooth functions with bounded derivatives. Suppose that,
for any xo E W, there exist io and ro E ( O, 6 Jco) such that for any j withxo E μj^1 (0, oo) we have Fj (xo) EB (Fio (xo), !ro) and inj (q) > 3ro for all
q E B (Fio (xo), ro). Then there is a function F : W----* M defined uniquely
by minimizing! I:?=i μi (x) d^2 (F (x), Fi (x)). In particular, F (x) satisfies
k
(4.26) L μi (x) expJ;(x) Fi (x) = 0.
i=l