1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

470 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

Then/ (0) = ao (t) = 'Ps,t (x) and/ (p) = ap (t) = 'Ps,t (y). Clearly

L (1) ?. ds(p,t) ('Ps,t (x), 'Ps,t (y)) ·
We shall estimate L (I) from above. Let Ju be the Jacobi field along au
with Ju (0) = 0 and Ju (s) = fi (u). Then
~(u)=Ju(t).
Since (Mn, g) has nonnegative sectional curvature, by the Rauch comparison
theorem (see Chapter 1 of Cheeger and Ebin [30]), we have

.,--1 Ju_( t---,-) I <!

I Ju (s)i - s.

Hence

~ [P

ds(p,t) ('Ps,t (x), 'Ps,t (y)) :SL (I)= Jo I~ (u)I du

:S ~ 1P IJu (s)i du=~ lap jfi (u)I du

t t ~

= ~ L (/3) = ~ds(p,s) (x, y).

This completes the proof of the proposition. D

An equivalent way to see that 'Ps,t in (I.16) is distance nonincreasing for

s :::; t < inj (p) is as follows.

SECOND PROOF OF PROPOSITION l.29 - VIA HESSIAN COMPARISON.

Given x ES (p, s) and Uo E TxS (p, s), let

/3: (-c:,c:)-+ S(p,s)

be a path with fi (0) = Uo. For u E (-c:, c:), define

au: [O,t]-+ M

to be the unique unit speed minimal geodesic with au (0) = p and au (s) =
/3 (u). Let

(I.18)

8

and v = av au ( v) '

so that Y'uV - \i'vU = [U, VJ = 0. By the Gauss lemma we have V =

\i'r, where r ~ d (-,p). Furthermore, Uo = U (ao (s)) and ('Ps,t)* (Uo) =

U (ao (t)). We compute

(I.19)! IUl^2 (ao ( v)) = 2 (\i'vU, U) = 2 (\i'uV, U) = 2 (\7\i'r, U &;; U)

since V = \7 r. Because ( M, g) has nonnegative sectional curvature, by the

Hessian comparison theorem,

(I.20)

1

\l\i'r:::; -g,

r