1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

8. CHARACTERIZING ROUND SOLUTIONS 89

each 'Yt, is a minimizer for I ( ·; g(ti)). Consider the pointed sequence of solutions

(5^2 , gi (t) , pi ), t E (-oo,-Li^2 t i ), where

9i (t) ~ Li^2 g (ti+ LT t)

and Pi E 'Yt,. Since 0 < R 9 (x , t) :::; Con 52 x (-oo, -1], we have

(29.68) R 9 , (x , t) :::; CL; on 52 x (-oo, - Li^2 (t i + l)]

(note that - Li^2 (ti+ 1) -7 +oo). Furthermore, by Klingenberg's theorem, we have

(29.69) inj(5^2 , g(t)) ~ :c fort E (-oo, -1].

Hence

(29. 70)

By Hamilton's compactness theorem for the Ricci fl.ow, there exists a subsequence

{gi(t)} which converges in the C^00 Cheeger- Gromov sense to a so lution (M~,

g 00 (t) , p 00 ) , t E (-00,00). By (29.68) and (29.70), we have that g 00 (t) is flat and

inj (g 00 (0)) = oo. This implies that (M 00 , g 00 (t)) is the static Euclidean plane.

Now the isoperimetric constant at t ime ti is

(29.71) I( t i ) = L~(t, ) ('Yt;) (A +^1 ht,; g( ti)) + A =^1 ht,; g(ti)))

= Area;,(o)(5!(ti)) + Area;,(o)(5~(ti)),

where 'Yt; divides 52 into 5 i (ti) and 5 ~ (ti).

On the other hand, since 19 ,(o) ht,) = 1, since 'Yt, is a C^00 embedded loop with

constant geodesic curvature with respect to gi(O), and since Pi E 'Yt,, we have that

under the Cheeger- Gromov convergence of gi(O) to g 00 (0) the sequence of loops 'Yt,

subconverges to some smooth embedded loop -y 00 , where L 9 =(o) ('Y 00 ) = 1.^2 Note

that by (29.60) we have that lig;(O) ('Yt;) is uniformly bounded. The loop 'Yeo divides
M 00 into two regions, one of which is bounded, which we call Meo,+, and one
of which is unbounded, which we call Meo,- · Without loss of generality, we may
assume that
Area 9 ,(o)(5!(ti )):::; Area 9 ,(o)(5~(ti)).

Then, by this and the convergence of 'Yt, to 'Yeo, we have that

and

Therefore, by applying this to (29.71), we conclude that I(ti ) ~ c for some constant

c > 0. (Actually, limi-+co I(ti ) = 47r.) This contradiction to (29.65) completes the

proof of the lower bound. 0

(^2) As in the proof of Lemma 29.18(4), one first observes that 'Yoo is an immersed closed curve
with a t most self-tange ncies and then shows that 'Yoo is embedded.