1547671870-The_Ricci_Flow__Chow

166 5. THE RICCI FLOW ON SURFACES

For the remainder of this section, we shall adopt notation different from
that used in the rest of this chapter. Namely, we will denote by ( M^2 , g ( t))
the solution of the Ricci flow
[)

otg = -Rg, g (0) =go

and by (M^2 , g ([)) the corresponding solution of the normalized Ricci flow

[) -

at 9 = -Rg, g (0) =go,

which exists for all t E [O, oo) by Proposition 5.19. As we shall see in Section
9 of Chapter 6, these are related by rescaling time

t = -r= (1 -e- rt) , t =! log _l_

f 1-ft

and dilating space

(5.52) g ( t) = e-rt g ([) = (1 - ft) g ([).

Here f (f) = f (0) denotes the average scalar curvature of the metric go.

Notice that g (t) exists for 0 ::::; t < T, where

(5.53)

If A (t) ~Area (M^2 , g (t)), the Gauss-Bonnet formula

87r = 47rX (M^2 ) = { R [g (t)] dμ [g (t)]

)M2

implies in particular that dA/dt = -87r and fA (0) = 87r. It follows that

(5.54) A ( t) = A ( 0) - 87rt = A ( 0) ( 1 - ft) ,

hence that A (t) ""'0 and Rmax (t) / oo as t / T, where Rmax (t) denotes
the maximum scalar curvature at time t.

The first step in the current strategy is to show that one can bound the
injectivity radius by the isoperimetric constant. This fact is a consequence
of Klingenberg's result (Lemma B.34) that inj (Mn, g) is no smaller than the
minimum of 7r / ../ K max and half the length of the shortest closed geodesic,
where Kmax denotes the maximum sectional curvature.

LEMMA 5.94. If (M^2 , g) is a topological 2-sphere with isoperimetric con-

stant CH (M^2 , g) and maximum Gaussian curvature Kmax, then

PROOF. Let "I be a geodesic loop of length L ("I) dividing M^2 into two

topological discs Mr and M§ with aMr = aM§ ="I· For i = 1, 2, let Ai