1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

208 5. ENERGY, MONOTONICITY, AND BREATHERS

so that

JM w^2 dμ 91 Q(g2, w) - JM w^2 dμ 92 Q(g1, w)

'.S 4 ( (1 + c;):g:+l - 1) JM w^2 dμ 91 JM IV'wl~ 1 dμg1

• 4 ( 1-(1 + c;)-n/

2

) JM w^2 dμ 91 JM IV'wl~ 1 dμg 1

• JM w

2

dμ^91 JM w

2

(I ( R^92 - R^91 ) ~~:: I + I ( ~~:: -1) R^91 I) dμ^91

• IJM w

2

( 1-~~::) dμg111/M Rg1 w

2

dμ^91 I·

(In the above estimates we took into account that R may change sign.) Let

fJ ~max { (1 + c;rl^2 - 1, 1 - (1 + c;)-n/^2 }, so that fJ---+ 0 as c;---+ 0. Since

1

1 - dμdμgl^92 I < - fJ ' we have

Hence

Q(g2, w) Q(g1, w)

JM w2dμg2 JM w2dμg1

'.S 4 ((l + c;):g:+i _ (l + c;)-n/2) JM l\7wJ~1 dμg1

JM w dμg2

+ ( (1 + fJ) max IRg2 - Rg1 I + 26 max IRg1 I) (1 + E: r1^2.

Taking w to be a minimizer for Q (g1, ·), we have

,\ (g2) - ,\ (g1)

'.S 4 ( (l + c;):g:+i _ (l + c;)-n/2) (l + c;r12 JM IV'wJ~1 dμg1

JM w dμg1

+ ((1 + fJ) max IR 92 - R 91 I+ 2fJmax IR 91 I) (1 + c;r/^2.