1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. EXISTENCE AND CONVERGENCE 93

On the other hand, applying (2.85) to bound the LP-norm of Mv(2) - wv,

we have for each v :2: 2,

1

( R;. L(R) (Mv(2) - Wv (y))Pdy)'

(2.90) :SC ( Mv(2) -Mv(l) +R

2

(q;n) II Hess(h)(rv,rv)llLq(B(2R))) ·

Combining (2.89) and (2.90), we have

(2.91)

1

(R~ n jB(R) r (w1 (y) - m1(2))Pdy) P

:SC (max (Mv(2) - Mv(l)) + RllY'hllco + R

2

<q;n) llY'VhllLq).

v::'.::2

Now let

w(sR) ~ t wv(sR) ~ t (sup Wv - inf Wv)

v=l v=l B(sR) B(sR)

N

= L (Mv (s) - mv (s)).

v=l

We then have

1

( R;n L(R) (w1(Y) - m, (2))")'

(2.92) :SC ( w(2R) - w(R) + RllY'hllco + R

2

<q-;_nl llY'VhllLq).

On the other hand, from (2.85) we also have

1

( R;n L(R) (M1 (2) - w1(y))P)'

(2.93) :SC ( w(2R) - w(R) + R

2

<q-;_nl llY'VhllLq).