1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. CONVERGENCE AT ALL TIMES FROM CONVERGENCE AT ONE TIME 135

So as tensors, we find that

Thus

I :t (rk -r)lk::; 3 l\7k (Rck)lk::; 3Jn - lCf.

3Jn - lCf lti - tol 2::: 1:

1

I :t (rk (t) - r) [k dt

2::: 11:1 :t (rk (t) - r) dt[k

2::: Irk (ti) - rik - irk (to) - r1k.

Hence we have a bound

irk (t) - rik ::; 3Jn - lCi It - tol +Irk (to) - rik

(3.10) ::; 3Jn"=1Ci It - tol + ~c^3 l^2 c1

using (3.8) and (3.5). Since It - tol ::; 'ljJ - (3, we have in (3.3): B (t, t 0 ) ::;

B ( 'ljJ, (3) for all t E [(3, 'l/J]. Thus by (3.9) and (3.10),

(3.11) l\7gk (t)I::; B (t, to)^312 l\7gk (t)lk::; C1,o,

where

61,0 =:= B^312 ( 'l/J, (3) ( 6Jn-=1Ci ( 'l/J - f3) + 3c^3!^2 c1).

This proves (3.4) for p = 1 and q = 0.

Next we prove inductively that for p 2::: 1,

I \7P Re k I ::; c; I \7P 9k I + c;' and I \7P 9k I ::; Cp,O

(where c;, c;', and Cp,O are independent of k). If p = 1, then using (3.8)

and (3.10),

l\7 Re kl (t) ::; B (t, to)^312 l(\7 - \7 k) Re k + \7k Re klk

::; B (t, to)^3 /^2 (Ir - rklk IRcklk + l\7k Rcklk)

::; B (t, to)^3 /^2 ( ( 3Jn-=1Ci l'l/J - f31 + ~c^3 /^2 C1) Cb+ Ci).

If the estimates hold for p < N with N 2::: 2, then we will prove them for
p = N. First we have

N

l\7NRckl = L\7N-i (\7 - \7k) \7t-^1 Rck + \7f Rck

i=l

N

::; L l\7N-i (\7 - \7k) \71-1 Rcki + l\7f Rcki.

i=l