1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

134 3. THE COMPACTNESS THEOREM FOR RICCI FLOW

where C ~ 2Vn=1Cb.

Now we compute

Clt1-tol 2: 1:

1

1:tloggk(t)(V,V),dt

2: 11:

1

! loggk (t) (V, V) dt/

= llog 9k (t1) (V, V) I

9k (to) (V, V) '

or equivalently,

e-G[ti -tolgk (to) (V, V) :S 9k (t1) (V, V) :S e^6 lti -tolgk (to) (V, V).

Hence we have

c-1e-6it1 -to lg (V, V) :::; gk (ti) (V, V) :::; Ce6it1 -tolg (V, V).

This completes the proof of (3.3).

For the second part we need to estimate the space-and time-derivatives

of 9k (t). We begin with estimating the first-order covariant derivatives of

9k (t). Note that

a d d

\7 a (gk)bc = axa (gkhc - r ab (gk)dc - r ac (gkhd'

so if we take the right combination, we see that

(gkrc (\7 a (gk)bc +\lb (gk)ac - \7 c (gk)ab)

= 2 (rk)~b - r~b - (gkrc r~c (gk)bd - r~b

• (gk)ecric (gk)ad + (gkrcr~b (gk)ad + (gk)ecr~c (gkhd
  (3.7) = 2 (rk)~b - 2r~b·
  This implies that

(3.8)

From

we have

(3.9)

Hence the tensors \7 9k ( t) and r k ( t) - r are equivalent.
We recall that the derivative of the Christoffel symbols (see, for instance,
(6.1) on p. 175 of Volume One) is

:t (I'k)~b = - (gktd [(\lk)a (Rck)bd + (\lk)b (Rck)ad - (\lk)d (Rck)abl ·