1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

8. CHARACTERIZING ROUND SOLUTIONS 95

for 0 :::; r1 :::; r2 :::; r3 :::; r4. Hence, for r 2 2,

Areag(tJ(B gCtl(r-1)) 2 (r ~rl)

2

( Areag(tJ(BgCtl(r + 1)) - Areag(tJ(BgCtl(r - 1))).

Let p E S^2. It suffices to prove the lemma for r :::; ~ diam(g( t)). Choose x E

S^2 with dg(t) (x ,p) = r 2 2. Then B$(t) (1) C BJ,(t) (r + 1) - BJ,(t) (r - 1) and

BJ,(t) (r - 1) c B$(t) (2r - 1), so we obtain Ya u 's "at least linear volume growth"

estimate:

Area 9 (t)(B $(t) (2r - 1)) 2 ;

6

Areag(t)(B$(t) (1)).

Since 0 < R:::; Con S^2 x (-oo, -1] and by the injectivity radius estimate (29.69),

this implies that Areag(tJ(B$(t)(l)) 2 c for some positive constant c. We conclude

that Area 9 (t)(B$(t)(r)) 2 er for all p E S^2 , r 2 3, and t E (-oo, -1]. 0

Now we are ready to prove that the lower bound for the areas of balls in Lemma

29.27 can be expressed in terms of time instead of radius. By a parabolic rescaling

of the solution g(t), we may assume without loss of generality that t he S^1 (r=) in

Lemma 29.24 and throughout t his discussion has radius l.

R ecall that 'Yt denotes a minimizer of I ( · ; g ( t)). Choose points xt E Si ("It) so

that

(29.81)

Since (S^2 , g(t), Pt) sequentially converges as t-+ -oo to cylinders (~xS1, g=(O),p=),
we have limt-+-= Pt = oo. (In the case of t he King- Rosenau solution, "It is the
equator and xt are the poles.)

LEMMA 29 .28 (Area bounds in terms of time). There exist T 0 < 0 and c 0 > 0

such that

(29.82)

for all t:::; To.

PROOF. Since Pt = dg(t) (xt, "ft) by definition and since L 9 (t) bt) is uniformly

b ounded by (29.66), there exists a constant C < oo such that for all t E (-oo, -1]

we have that

(29.83)

On the other hand, s ince t he distance circle 8Bx,^9 ~)(pt) is connected (and near

"It) by Lemma 29.25, we have that

Area(B!~)(Pt + C) - B!~)(pt)):::; C

t t

independent oft. Hence, by (29.83), we have

(29.84) A±bt):::; Area(Bx t^9 ~)(Pt + C)):::; Area(Bxt^9 ~l(pt)) + C.

The lemma now follows from Lemma 29.23. D

We may also bound t he distance Pt by a constant times it!.