1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

486 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION

We have

(I.49)

Claim. If "(EN/ (H), where 0 < c::::; l (A, K,), then

(I.50) (i) :s II'tl (0, t) 2: -AcL (7r o "!),

(I.51) (ii) :: 2 II'tl (s, t) 2: -E^2 K,^2 II'tl (s, t)

for all s and t.

Proof of claim. (i) Using 'VrsI't-Y'rtrs = L [l, gt] = 0, we compute

0 1 0 2

II'tl OS II'tl = 2 OS II'tl

= (Y'rsI't, I't)

= (Y'rtrs, I't)

0

= ot (rs, rt) - (rs, V'rtrt).

Ats= 0 we have II'tl = L(7ro"f) and (I.47), which implies gt (rs,I't) = 0
at s = 0. Hence, using (I.48), we have

8

L (7r 0 "!)as II'tl (0, t) = -d (7r 0 "( (t) '"( (t)) (v (7r 0 "( (t)) 'Y'rtI't) (0, t)

= d ( 7f a "! ( t) , "! ( t)) II o (rt ( o, t) , rt ( o, t)).

Since II o =UH 2: -AIH and II't (0, t)I = L (7r o "f), we have

0

OS II'tl (0, t) 2: -Ad (7r 0 "( (t) '"( (t)) L (7r 0 "!)

2: -AcL (7r o "!)

because d (7r o "! (t), "! (t)) ::::; c by assumption.

(ii) We compute

02 1 a2 2 ( 0 )2

II'tl as2 II'tl = 2 os 2 II'tl - os II'tl

(I.52) = (V'rs V'rsrt, rt)+ IY'rsrtl^2 - (a as II'tl )2

Since

IY'r.I'tl

2

• (! II'tl)

2

= II'~l 2 (IY'rsI'tl

2

II'tl

2

• (V'rsI't, I't)

2

) 2: 0

and since