1547671870-The_Ricci_Flow__Chow

MONOTONICITY OF THE ISOPERIMETRIC CONSTANT i59

For E > 0 small enough, '"Ye is a smooth embedded loop. Since g is Euclidean

up to first order at x, we see from (5.47) that as E "\i 0,

D

We now sketch a proof of the following existence result; further details
may be found in §3 of [62].

LEMMA 5.85. ff ( M^2 , g) is a Riemannian surface diffeomorphic to S^2

such that CH (M^2 , g) < 41T, then th ere exists a smooth embedded loop (3

such that

Our proof of Lemma 5.85 will use the following algebraic fact.

LEMMA 5.86. If Li, L2 and Ai, A2, A3 are any positive numbers, then

2 ( 1 1 ) (Li+L^2 ) Ai+A2 + A3

mm ·{2(1 Li Ai + A2 1) + A3 '

PROOF. To obtain a contradiction, suppose the result is false. Then there exist positive numbers Li, L2 and Ai, A2, A3 such that

and

(Li+ £^2 )

2

(Ai! A2 + ~3) '.SL~ (~2 +Ai! A 3). We rewrite these inequalities as

~~~~-Ai (A2 + A 3) < Lr 2 A 3 (Ai + A2) - (Li + L2)

(5.48)

and

(5.49)

A2 (Ai+ A 3) L~

A3 (Ai + A2) '.S (Li + £ 2 )^2.

Summing (5.48) and (5.49) yields

which is impossible. D