1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

478 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

and

(I.35)

We shall prove a uniform positive lower bound for ¢i· In this regard,

without loss of generality, we may assume ef>i :::; ~ (so that cos ef>i 2::: 0).

Case (1). Suppose Bi 2::: ~· By (I.33) and (I.35), we then have

sin ei > sin c.

In the Euclidean comparison triangle, by the law of sines,

. - di. - di.
sm¢i = -( ) sm(h > d smc
r x i + ri

since r (x):::; di+ n. Hence, by (I.31), we have sinef>i > c:sinc, so that

ef>i 2::: sin -l ( c: sin c)

since ef>i :::; ~·

Case (2). Now suppose Bi< l By the Euclidean law of cosines applied

to Li, we have

so that

rir (x) cos ef>i = ~ (rf + r^2 (x) - df)

< - r~ i

since Bi < ~ implies r^2 ( x) :::; rf' + df. Hence

• ri
  cos ¢i :::; r ( x) :::; 1 - c:,

where we used (I.30).

We conclude from Cases (1) and (2) that

(I.36) L (fi (0), 'YVi (0)) = </>i 2::: ef>i 2::: 5,

where

5 =min { sin-^1 (c: sine), cos-^1 (1 - c:)} > 0.

Now, by passing to a subsequence, we may assume ri+l ;::: 2ri for all i.

By (I.36) with c: =!,we then have for any j of. i,^9

L (')iv; ( 0) , 'YVi ( 0)) ;::: sin -l ( si; c).

This is a contradiction to the compactness of the unit sphere s;-^1 and hence
the lemma is proved. D

The following consequence of the Toponogov comparison theorem is due
to Abresch [l].

(^9) Note that cos- (^1) ~ = ~;:::: sin- (^1) (si~c).