1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

426 H. CONVEX FUNCTIONS ON RIEMANNIAN MANIFOLDS

(3) (B (p, 2r) , g) is lOO~n 2 -close to the Euclidean ball (B (0, 2r), 9Euc)

in the C^2 -Cheeger-Gromov topology.

Choose an orthonormal basis {e1, ... , en} of TpM· By assumption, for

each i = 1, ... , n,

s f---1-f o expP (sei)

is a convex function of s E (-2r, 2r). It is well known that f o expP (sei) is

a continuous function of s and that f o expP (sei) is bounded from above by

max {f o expP (-rei), f o expP (rei)} on the interval [-r, r].

Let k E N with 2 :::::; k :::::; n. Given a sequence 1 :::::; ii < · · · < ik :::::; n and

a sequence of k signs±, ... ,±, we define the (k - 1)-simplex

.6. :f:, ii, ... ... ,:;1: ,ik c B ( , p 2r)

inductively as follows. We take the sequence of signs to be -, ... , - as an

example. Define the 1-simplex

to be the minimal geodesic from expP (-reik_ 1 ) to expP (-reik).
We then define the 2-simplex

to be the smooth surface (with piecewise smooth boundary) spanned by

all minimal geodesics from expP (-reik_ 2 ) to some point in ~4,:,ik' and so

forth. At the last stage, we obtain the (k - 1)-simplex

In general, taking k = n, we have 2n smooth ( n - 1 )-simplices

.6. ±, ... ,±

1, ... ,n

corresponding to the different choices of sign.

From the convexity of s f---t f ( expq ( s V)) and from the construction of

.6.t,:::;;=, it is easy to see that JIA±, ... ,± is bounded by the largest value off

on th e ver tices. o f .w.A±, 1 , ... ... ,n ,± , i.e., •

1, ... ,n

!IA±, l, ... ... ,n ,±

:::::; max {f oexpP (-re1),Joexpp (re1), ... , f oexpP (-ren),JoexpP (ren)}.

Now we define an 'n-body'

consisting of minimal geodesics from p to some point in any one of .6.t,'.::;~±.
Again, by the convexity off, we have JIA is bounded by

max {f (p),f oexpp (-rei),J oexpp (re1), ... , f oexpp (-ren)J oexpp (ren)}.