- ISOPERIMETRIC CONSTANT OF METRICS ON S^2 85
The curve shortening flow is the case where f N = -r;,N is the curvature vector
of f3t·Let li(t) be the solution to the curve shortening fl.ow with li(O) = Ii· Let
Li(t) = L(li(t)) and A±,i(t) = A±hi(t)). By (29.57), we have
(29.58a) -dLi = -^1 r;,^2 ds < 0
dt ,,(t) - '(29.58b) d Ad±,i = =t= 1 r;,ds = _21 { R9dμ9 - 27r E [-C, CJ,
t ,,(t) Js'i(r;(t))
where the last equ ality is by the Gauss-Bonnet formula and since Sf (li(t)) is an
embedded disk. Here, we h ave chosen the unit normal N to yield the given signs
for d~z".By differentiating (29.53), we have that as long as A±,i(t);::: g: > 0 holds,
- d l n I( Ii ( t )· , g ) - 2 --dLi - -----^1 dA+,i - -----^1 dA_,i
dt Li dt A+,i dt A-,i dt
< --^21 r;,^2 ds + C
- c 7;(t)
since Li(t) ::::; Li(O) ::::; C. Let Co = C^2 /2. For each i , suppose that the solution
li(t) to the curve shortening fl.ow exists on a maximal time interval [O, Ti)· Let T;
be the maximum time for which A±,i(t) 2: g: holds fort E [O, T;).
Claim. There exist ti E [O, T;) such that li(ti) is a minimizing sequence withL(li(ti)) > 6 and
(29.59) 1 r;,^2 ds ::::; C 0.
7;(t;)Suppose that J,,(t) r;,^2 ds > Co for all t E [O, T;). We then have the follow-
ing inequalities for all t E [O, T;). The isoperimetric constant is nonincreasing:{ft I(li(t); g) ::::; 0, so that I(li(t); g) < 47r - c:. Thus Li(t) > 6, which implies that
A±,i(t) > g:. Hence T; =Ti. Now, by Grayson's theorem for the curve shortening
fl.ow of embedded curves on surfaces (see [123] and [112]), since li(t) cannot shrink
to a round point, we conclude that li(t) converges to a closed embedded geodesic as
t ---+ oo. In particular, limt--.oo J,,(t) r;,^2 ds = 0, which is a contradiction. Therefore
there exists first time t i E [O, T;) such that J,,(t,) r;,^2 ds ::::; Co. Then L(li(ti)) > 6
and I(li(ti)i g) ::::; I(li; g), so that bi(ti)} is a minimizing sequence. This proves
the claim.
Let Ii ~ Ii (ti), let { x"'} be local coordinates on S^2 , and let If = x°' o Ii. Let r;,i
and Ni denote the geodesic curvature and unit normal of Ii, respectively. Assume
that the metric components and Christoffel symbols satisfy c-^1 5af3 ::::; 9 af3 ::::; C6af3
and lf'p 6 i ::::; C , so that 2::~= 1 ( dJ}" r ::::; C. We have
a Ii Ii -a Ii Ii
(
d
)"' d2°' df3do
-r;,iNi = 'V*ds = ds2 +rf3odsds'so that