1547671870-The_Ricci_Flow__Chow

7. HIGHER DERIVATIVE ESTIMATES AND LONG-TIME EXISTENCE 205

the metrics g (t) on the time interval [O, T). To estimate the first derivatives
of the metric, we begin by writing

fft (a~ig.ik) = a~i (fftg.ik) = -^2 a~iR.ik

= -2 ( 'ViRjk + rfjRRk + rfkR.iR).

Because IRm (x, t)l 9 :::; K by hypothesis, this implies that

(6.49) lfftagl = 2laRcl:::; 2IV'Rcl +CKlfl.

By equation (6.1), the tensor fftr is given by

whence we get the estimate

Now by Corollary 6.47, there exists B = B (m, n, K, /3) such that the bound

IV' Rel :::; B holds on the time interval (/3/ K, T). Since lfl is bounded on

[O, /3/ K] by some A = A (K, {3, go), we see that

(6.50) If (x, t)I:::; A+ BC (T - {3/ K):::; C

for all x E Mn and t E [O,T). Since IV'Rcl is bounded on [0,/3/K] by some

D = D (K, {3, go), we conclude by (6.49) that

lagl :::; Iago I+ CD;+ (2B + C) (T - {3/ K) :::; C.

To do the general (inductive) step, let a = (a1, ... ,ar) be any multi-

index with la l = m. Then since

a (alal ) (alal )

at ax°'gij = -^2 ax°'~.i '

it will suffice to bound lalal Rel. A moment's reflection reveals that we can

estimate

m m-1

(6.51) lam Rel:::; I:Ci lril l'Vm-iRcl + L c~ lair! lam-l-iRcl ,

i=O i=l