1547671870-The_Ricci_Flow__Chow

162 5. THE RICCI FLOW ON SURFACES

We are now ready to state the main assertion of this section. Notice that
it does not matter whether we consider the normalized or unnormalized Ricci
fl.ow, because the isoperimetric constant is scale invariant:

Cs (M^2 ,g) =Cs (M^2 ,e>-g).

THEOREM 5.88. Let (M^2 , g (t)) be a solution of the Ricci flow on a

topological 2-sphere. At any time t such that CH (M^2 ,g (t)) < 47f, one has

:tCH(M^2 ,g(t)) 2:0

in the sense of the lim inf of forward difference quotients.

REMARK 5.89. Because CH (M^2 , g (t)) is a continuous function of time

satisfying the upper bound CH (M^2 ,g (t)) :::; 47r, the theorem implies in

particular that CH ( M^2 , g ( t)) is nondecreasing in time.

Before proving the theorem, we introduce some notation. Assume in the

remainder of this section that (M^2 , g ( t)) is a solution of the Ricci fl.ow on

a topological 2-sphere for t E [O, w). Let

A= A (t) ~ A 9 (t) (M^2 ).

Observe that it follows from Lemma 6.5 and the Gauss-Bonnet Theorem

for a closed surface, that

(5.50) ~A= - { RdA = -47rX (M^2 ) = -87r.

dt }M2

Given any to E [O, w) and any smooth embedded loop /'o separating M^2

into Mt and M 0 , let /'p denote the parallel curve of signed distance p from

/'o measured with respect to the metric g (to), where p < 0 for curves in

Mt and p > 0 for curves in M 0. For IPI sufficiently small, /'p is a smooth

embedded loop that separates M^2 into Mt and Mp-. We shall thus consider

the following functions of both p and t:

L = L (p, t) ~ L 9 (t) (/'p)

A±= A± (p, t) ~ A 9 (t) (M~)

_. 2 A (t)

CH=CH(p,t) 7 (L(p,t)) A + ( p, t )·A - ( p, t )"

Notice that by (5.47), CH is the isoperimetric ratio of the curve "Ip taken

with respect to the metric g (t). The next three lemmas accomplish the

calculations necessary to prove the theorem.

LEMMA 5.90. Let (M^2 ,g(t))

logical 2-sphere. Then

ff PL= f'Yp k ds

ffpA± = ±L

be a solution of the Ricci flow on a topo-

2-L at = -f 'Yr Kds