88 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS
Since the function t HI (t) may not be differentiable, given a time f, we define
ddt^1 (f) ~ lim inf vr I(t~=~([) to be the lower converse Dini derivative. By
applying the first and second derivative test to Lemma 29.20, we have that the
isoperimetric constant worsens at least at a certain rate going backward in time.
PROPOSITION 29. 21 (Decay of the isoperimetric constant).
( 1) The isoperimetric constant I ( t) of a nonround solution g ( t) satisfies(29.64) d_ dt I (t) -> I(t) 47r 47r - I ( ltl t) JOr^1 t E ( - oo, 0 ).
In particular, the function t H I ( t) is strictly increasing.
(2) For any t 1 E (-oo, -1] and all t E (-oo, ti] we have that(29.65) I ( ) 47r
t :::; 1 + c ltl'. 47r-l(t1)
where c 7 1 (ti) ltil.PROOF. (1) By Lemma 29.18(4) and since g (t) is not a round shrinking 2-sphere, we have that I (t) < 47r for all t E (-oo, 0). Then by Lemma 29.18(2),
given any time to E (-oo, 0) , there exists a smooth embedded closed loop 10 such
that I (lo; g(to)) = I(to). Since % I lnl(p, to)= 0 and ;2 2 I lnl(p, to) :'.'.'. 0, by
P p=O P p=O
Lemma 29.20 we have
_1_ ~1 I (O, t) > 47r - 1(0, to).
1(0, to) at t=to - 47r ltolSince I (t) :::; I (0, t) fort:::; t 0 and I(t 0 ) =I (0, t 0 ) , we obtain (29.64).
(2) Let J (t) ~I (t) - 1 ~~ltl, so that J (t 1 ) = 0. From part (1) we have thatd_ J t > -J (t)
2
~ c lt l - 1 J t
dt ( ) - 47r ltl + ltl c ltl + 1 ( ).Inequality (29.65) now follows from Corollary 10.26 in Part II. D
Let /t be any C^00 embedded loop minimizing I ( · ; g( t)); we estimate its length.
PROPOSITION 29.22 bt is uniformly bounded from above and below). There
exists a constant CE [1, oo) such that fort E (-oo,-1],(29.66)PROOF. 1. Upper bound. By Proposition 29.21(2), for t E (-oo, -1] we have
that
(29.67)where the areas A± bt) are measured with respect to g(t).
- Lower bound. The proof is by contradiction. Suppose that there exists