1547671870-The_Ricci_Flow__Chow

230 7. DERIVATIVE ESTIMATES

where f3m is a constant to be chosen momentarily. We will explain the full
strategy behind this choice after we estimate the evolution equation satisfied
by G. For now, we simply observe that our assumption on IRml implies that
G:::; f3m (m - 1)! K^2 at t = 0. By (7.4) and the inductive hypothesis, there
are constants CA depending only on k and n such that for .all 1 :::; k < m we
have

(7.6) :t IV7k Rml2 :::; ~ IV7k Rml2 - 2 IV7k+l Rml2 + c~~3

on t he time interval 0 < t :::; a/ K. It will be significant that we retain the

good term - 2 jV7k+^1 Rmj

2

that appears in the inequality (7.6) for k < m,

although we d iscarded it in estimate (7.5). Indeed, using formulas (7.5) and

(7.6), we compute that G satisfies the differential inequality

:t G :S ~G + CmKtm IY7m Rml^2 + CmK^3 + mtm-l IY7m Rml^2

m { - 2tm-k jV7m- k+l Rmj

2

(m - 1)! + C-m-k K^3 }

+ f3m L ( m - k) 1 •

k=l. + (m - k) tm-k-l jV7m- k Rmj2

This reveals why we defined G as we did: the good terms

-2 (m - 1)! tm-k IY7m- k+l Rml2

(m - k)!

that we get when we differentiate jV7m-k Rmj^2 compensate for the bad terms

_(m_-_1)_! _ (m - k + 1) tm-k IV7m- k+l Rml2

(m - k + 1)!

we got when we differentiated tm- (k-l). In this way, we get the estimate

:t G :S ~G + (CmKt + m - 2f3m) tm-^1 IV7m Rml^2 +(Cm+ f3C:n) K^3 ,

where

-,. ~ (m - 1)! -

Cm=:= L.,, (m _ k)! Cm-k·

k=l

Choose f3m ~ ( Cma + m) /2, noting that f3m only depends on m, n, and
max {a, 1 }. Then for t E [O, a/ K], we have

:t G:::; ~G + (Cm+ f3C:n) K^3.

Since G :S f3m (m - 1)! K^2 at t = 0, the maximum principle implies that

sup G (x, t) :::; f3m (m - 1)! K^2 + (Cm+ (3C~) K^3 t:::; C!K^2

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