1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. CENTER OF MASS AND NONLINEAR AVERAGES 175

implies J/;^1 (13) is closed and bounded, and hence compact since 9k is com-

plete. Therefore B is the image of a compact set, and hence compact. We are

done because if closed metric balls are compact, then the metric is complete

(see, for instance, [72]). D

5. Center of mass and nonlinear averages

In this section we review some standard work on the convexity of the

distance function and properties of the center of mass. The treatment here

follows mostly that in Buser and Karcher [40], although we address addi-

tional issues related to proving C^00 -convergence. In this section we adopt

the convention 7r / ( 2.JK) ~ oo when K :S 0 and we assume that geodesics

have constant speed, i.e., they are parametrized proportional to arc length.

5.1. Derivatives of the distance function and exp-^1 • Let (Mn,g)

be a Riemannian manifold and let d (x, y) denote the distance between x

and y. Fix x E M and consider the function f (y) ~ ~d^2 (x, y). We can

write fas an integral by letting I (r) be a minimal geodesic from I (0) = x
tor (1) = y, so that

f (y) = ~ fo

1

g (";, ry) dr

since geodesics have constant speed. The quantity g ( ry, ry) is constant and

equal to the square of the length of the geodesic. The gradient of f can be

expressed as follows.

LEMMA 4.43 (Gradient of the distance squared function). If y is not in

the cut locus of x, then

gradf (y) = -exp;^1 x E TyM·

PROOF. Let r (r) be the unique minimal geodesic from r (0) = x to
r (1) = y; then by the Gauss lemma, grad f (y) = ry (1). It follows easily from
the uniqueness of solutions of the geodesic equation that expy (-ry (1)) = x,

so that -ry (1) = exp;^1 x. The lemma is proved. D

Given Y E TyM, let a : (-E, E) -t M be the geodesic with a (0) = y

and a (0) = Y (where E > 0 is sufficiently small for later purposes). Since y

is not in the cut locus of x, there exists a smooth family of unique minimal

geodesics rs : [O, 1] -t M, such that is (0) = x and rs (1) = a (s) for

s E (-E, E). Then lo = r is the minimal geodesic joining x and y. Define

CT: (-E,E) x [0,1]-tM by

CT(s,r) ~ls(r) =expx [rexp;^1 a(s)] =expa(s) ((1-r)exp:(s)x),