1547671870-The_Ricci_Flow__Chow

164 5. THE RICCI FLOW ON SURFACES

LEMMA 5.92. If (M^2 , g (t)) is a solution of the Ricci flow on a topological

2-sphere, the isoperimetric ratios CH (p, t) of the parallel loops '"'(p measured
with respect to the metric g (t) satisfy the heat-type equation

a 82 r a 4n - cH (A+ A_)

at (log CH)= 8p2 (log CH)+ L. op (log CH)+ A A_+ A+ '

where

r ~ 1 kds.

rp

PROOF. Notice that

log CH= 2logL - log A+ - log A_+ log A.

By Lemmas 5.90 and 5.91, we have

8 8

2

8

2

( 8 )

2

at (logL) = L-^1 op 2 L = op 2 (logL) + op (logL)

82 r a

= - (logL) + - · - (logL).

ap^2 L op

Similarly, we obtain

:t (log A±) = A±^1 [ :; 2 A± - 4n ± r]

=

82

(log A±)+ (~(log A±))

2

4

n ± _!:__

ap^2 op A± A±

8

2

( L )

2

= - (log A±) + - - - + -4n r · -^8 (log A±) ,

8p2 A± A± L op

because L = ± (8A±/8p). Finally. we recall (5.50) to get

d 8n

dt (log A)= -A.

The lemma follows when we combine these results with the identities

4n (A+ +A) = 4n (A++ A)

2

• 2A+A-= 4n + 4n 8n
  A A A+ A+ +A- A+A- A+ A_ A
  and

D

If CH < 4n, then Lemma 5.92 shows that log CH is a supersolution

of a type of heat equation. This leads one to expect that it should be

nondecreasing. We now prove this rigorously.