1547671870-The_Ricci_Flow__Chow

7. HIGHER DERIVATIVE ESTIMATES AND LONG-TIME EXISTENCE 207

Reasoning as in the derivation of (6.51) with the tensor Re replaced by \7 Re,
we observe that

i-1

loi\7Rcl s I:cj lrjl 1vH^1 -jRcl + I:cj 1ajrl 1ai-jRcl,

j=O j=l

where the constants Cj, cj depend only on p and n. Applying this estimate

to (6.52) then shows that

where all of the terms on the right-hand side have already been bounded.
Thus we have I gtam-^1 r1 '.SC and hence 1am-^1 r1 '.S C+CT. This completes
the proof. D

An examination of the proof makes it evident that we have also estab-

lished the following result.

COROLLARY 6.51. Let (Mn,g(t)) be a solution of the Ricci flow on a

compact manifold with a fixed background metric g and connection 'V'. If

there exists K > 0 such that

[Rm (x, t)[ 9 '.SK for all x E Mn and t E [O, T),

then there exists for every m EN a constant c:.n_ depending on m, n, K, T ,

go, and the pair (g, '\7) such that

IVmRc (x, t)l 9 '.S c:n for all x E Mn and t E [O, T).

REMARK 6.52. As an alternative to the argument used to prove Propo-

sition 6.48, one may begin by writing the identity

in the form

a e e

oxigjk = rijgek + rikgje·

Then using the uniform bounds on g (t) given by Corollary 6.50, one gets

the estimate

and proceeds as before.