1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

(^204) 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

Regarding the f1h term above, we have (using \7 H = -H \7 J)

r f1h ~ r

JM hH dμg(O) =JM (t1h)H dμg(O)

=JM (\7h, \7 f)H dμ 9 (o)

=JM (\7 log h, \7 f)H dμ 9 (o)

(22.80) =JM (\71ogh, '7})iI dμg(O) +JM l\7loghJ^2 H dμg(O)·

Applying (22.80) to (22.79), we obtain

JM (2f1f - j\7 fJ

2

) fI dμg(O) =JM ( 2f1f - /\7 f/

2

• j\7 log hj^2 ) H dμg(O)·

Hence (22. 78) yields

(22.81)

~l (1- 10~2)

:::;-JM (t( R+2f1f-/\7f/

2

-l\7loghJ^2 ) +f +logh-n) Hdμ 9 (o)

=JM (-t/\7f/

2

-1 + n) H dμg(O)

+JM (t(-Rh+ l\7:1

2

)-hlogh) Hdμ 9 (o),

where we used integration by parts.

In view of (22.81), the desired inequality (22.64) follows directly from

showing that

(22.82) r (-( l\7hl

2

JM t -Rh+-h-) -hlogh ) Hdμg(O)::::; A^1 2

2 +c.

STEP 2. A sufficient inequality to prove (22.82).
Since (22.50) and b = lOcoA, we have that h (·, 0) E [O, 1] vanishes
outside of Bg(O) (xo, 20coA). By the assumption that R;::::: -1 in B 9 (o)(x 0 , 1)
and 20coA::::; 1, we have

(22.83)

where we used f :=::; c;^2.