1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

468 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

when>.. is sufficiently small. This and (I.11) imply dGH (N;..,B;..2g (O,r)) '.S

~. Now we have for >.. small enough

dGH ( Bdcone(M(co)) ( 0' r) 'B~2g ( 0' r))

'.S dGH ( Bdcone(M(co)) (0, r), N) + dGH (N, N;..) + dGH (N;.., B;..2g (0, r))

'.S c:.

D

3. A monotonicity property of nonnegatively curved manifolds

within the injectivity radius

In this section we discuss a monotonicity property of distance spheres in

nonnegatively curved manifolds when their radii are less than the injectivity

radii of their centers. In particular, within the injectivity radii, there are

relatively distance-nonincreasing maps between distance spheres of differ-

ent radii centered at the same point which agree with the flow along rays

emanating from that point. The proof of this uses the Rauch comparison

theorem,^3 which says that geodesics emanating from a point spread out

slower than for the model Euclidean space. We may think of this result as

motivation for the existence theorem for asymptotic cones in the previous

section. ·

Given p EM, we have that any unit speed geodesic

f3: [O,inj (p))--+ M

with f3 ( 0) = p is minimal and the map

expP: B (o, inj (p)) --+ B (p, inj (p))

is a diff eomorphism.

Recall that S (p, s) denotes the distance sphere of radius s centered at

p. For any 0 < s '.S t < inj (p), define the map

(I.13) <p 8 ,t: S (p, s)--+ S (p, t)

by

(I.14)

where ~ : TpM --+ TpM denotes multiplication by ~.^4 Note that <?s,t is a

diffeomorphism between smooth spheres for 0 < s '.S t < inj (p).

As a subset of the metric space (M, dg), where dg denotes the metric

induced by the Riemannian metric g, for any s > 0 the sphere S (p, s) is

(^3) Alternatively, use the Hessian comparison theorem..
(^4) Note that 'Ps,t (a (s)) =a (t) for any (minimal) unit speed geodesic a: [O, t] _,. M
with a (0) = p.