1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

310 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS

REMARK 25.5. If n = 2, then under the assumptions of the above lemma

we have for any real number p > 2

(

1 /fl p~2 dμ_g) p;2

Bg(xo,r)

:::; eCs(P) ( l+VKr) Vol;~ B_g (xo, r) r (r^2 /\7 f I~ + f^2 ) dμ_g,

J Bg(xo,r)

where Cs (p) < oo depends only on p.

Now given xo E int (n), we take D = B_g (xo, r), where r E (0, 2ro], where

ro satisfies (25.8). We then have

supp ('If;) c B_g (xo, r) x [ri, r2].
Assume from now on that n 2:: 3 (we leave the n = 2 case as an exercise). In
view of (25.19), we apply the above £^2 -Sobolev inequality to the function
'lj;vP, which is possible since Rc_g 2:: -K in n. We obtain for any p E [1, oo)
and r E [ri, T2],
n-2

(

r /1f;vP/n

2

."..'2 dμ_g)---;n (r)

} Bg(xo,r)

:::; e^0 s(i+v'Rr) Vol§~ Bg (xo, r) ( r (r^2 /\7 ('lj;vP)/~ + '1j;^2 v^2 P) dμ_g) (r).

J Bg(xo,r)

On the other hand, by Holder's inequality, since n;:;-^2 + ~ = 1, we have

1

/1f;vP/_n_dμ_g 2(n+2) = l /1f;vP/^2 /1f;vP/n^4 dμ_g

B9(xo,r) B9(xo,r)

Thus

(25.23)

n-2 2

:::; (1 /'lj;vP/!."..'2 dμ_g\ ---;n( r /'lj;vP/2 dμ_g\ ~

B9(xo,r) J } Bg(xo,r) J