1547671870-The_Ricci_Flow__Chow

14. MONOTONICITY OF THE ISOPERIMETRIC CONSTANT 161

area 'outside' of 'Yoo· We then have

CH ('Yoo)= (L1 + L2)

2

(~

3

+ A

1

: A

2

)

CH (r1) =LI (~1 + A2: A3)

CH (r^2 ) = L~ (~2 + A1: A3).

But Lemma 5.86 implies that

CH boo)> min {CH (r1), CH (r2)},

which contradicts the minimality of 'Yoo· D

It is interesting to note that CH (r) may be represented in another way.

LEMMA 5.87. Let ( M^2 , g) be diffeomorphic to 52. Let 'Y be a smooth

loop in (M^2 ,g), and let (5^2 ,9) be a 2-sphere of constant curvature chosen

so that

Ag (5^2 ) =Ag (M^2 ).

Let ')' be a shortest loop (namely, a round circle) that separates ( 52 , 9) into

two discs Dr and D§ with Ag (DI) = Ag (MI) and Ag (D§) = Ag (M§).

Then

C (r) = 47r Lg (r)2.

H Lg (1')2

PROOF. Identify (5^2 , 9) with the 2-sphere of radius p ~ J Ag (M^2 ) /47r

centered at 0 E IR.^3. We may assume without loss of generality that

0 < A1 ~Ag (Mi) ::::; A2 ~Ag (M~).

Choose h E [O, p) such that

Then

Ag {(x,y,z) E 52 : z > h} = A1

Ag {(x,y,z) E 52 : z < h} = A2.

1 ~ { (x, y, z) E 52 : z = h}

is a round circle that divides ( 52 , 9) into two discs with areas A1 and A2.

Because

• d Ag {(x,y,z) E 5 2 : z < h} = dh d Jh 27rpdz = 27rp,
  dh - p
  we have A2 = 27rp (p + h) and A1 = 27rp (p - h). Hence

(^4) 7r A1 A1A2 + A2 - [27rp (p - h)] p 2 [27rp (p + h)] -^4 7r 2 ( P 2 h2) - (L-g (-))2 'Y.
The lemma follows. D