1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. CONVERGENCE AT ALL TIMES FROM CONVERGENCE AT ONE TIME 133

and the time-derivatives and covariant derivatives of the metrics 9k (t) with

respect to the metric g are uniformly bounded on K x [,B, 'ljJ], i.e., for each

(p, q) there is a constant Cp,q independent of k such that

(3.4)

for all k.

I

-\7P atq aq gk( t ) -I < C -p,q

REMARK 3.12. Since we often assume bounds on Rm whereas the metric
evolves by Re, we note

-Jn=l IRml g :S Re :S v'n -1 IRml g.
Since we often interchange g and 9k norms, we recall the following ele-
mentary fact.

LEMMA 3.13 (Norms of tensors with respect to equivalent metrics). Sup-
pose that the metrics g and h are equivalent:

c-^1 g :Sh :S Cg.

Then for any (p, q)-tensor T, we have

(3.5) ITlh :::::; c(p+q)/^2 1r1 9.

PROOF. We can diagonalize g and h so that 9ij = bij and hij = Aibij.
The assumption implies c-^1 :S Ai :SC for all i. Then

and

I

Tl2 = ~h ... h hi1j1 ... hiqjqy~1··:kpT~1 .. ·~p

h ~ k1£1 kp£p i1 ···iq Jl "'"Jq

PROOF OF LEMMA 3.11. For the first part, since

a

atgk (t) (V, V) = -2Rck (t) (V, V)

IRck (t) (V, V)I :S ~Cbgk (t) (V, V),

we can estimate the time-derivatives

I

~ 1 (t) (V V) I = 1-2 Re k (t) (V, V) I < 2Jn=lc'

at oggk ' 9k (t) (V, V) - n o·

We have proved

(3.6)

D