1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. BASIC RlCCI FLOW 463

So if go has positive Ricci curvature, then there exist constants C < oo and

8 > 0 such that

(A.23) ,\(Rm~~-~ (Rm) ~ C.

We shall call (A.23) the 'pinching improves' estimate.
Next we consider estimates for the derivatives of Rm.

2.5. Global derivative estimates. For solutions to the Ricci fl.ow on
a closed 3-manifold with positive Ricci curvature, we have the following
estimate for the gradient of the scalar curvature. (See Theorem 6.35 on
p. 194 of Volume One.)

THEOREM A.28 (3-dimensional gradient of scalar curvature estimate).

Let (M^3 , g (t)) be a solution of the Ricci flow on a closed 3-manifold with

g ( 0) = go. If Re (go) > 0, then there exist iJ, <5 > 0 depending only on go

such that for any (3 E [O, iJJ, there exists C depending only on (3 and go such

that

IVRl

2
< (JR-8/2 + c R-3.
R3 -
After a short time, the higher derivatives of the curvature are bounded
in terms of the space-time bound for the curvature. (See Theorem 7.1 on
pp. 223-224 of Volume One.)

THEOREM A.29 (Bernstein-Banda-Shi estimate). Let (Mn,g (t)) be a
solution of the Ricci flow for which the maximum principle applies to all the

quantities that we consider. (This is true in particular if M is compact.)

Then for each a > 0 and every m E N , there exists a constant C ( m, n, a)

depending only on m, and n, and max {a, 1} such that if

a

IRm (x, t)lg(t) ~ K for all x EM and t E [O, K],

then for all x EM and t E (0, ~],

I

(Vl-p. 224) \7 m Rm ( x, t )I g(t) ~ C(m,n,a)K tm/

2

With all of the above estimates and some more work, one obtains The-
orem A.26.
Finally we mention that an important local version of Theorem A.29 is
the following.

THEOREM A.30 (Shi-local first derivative estimate ). For any a > 0

there exists a constant C (n, K, r, a) depending only on K, r, a and n such

that if Mn is a manifold, p E M, and g (t), t E [0,T], 0 < T ~ a/K,

is a solution to the Ricci flow on an open neighborhood U of p containing

Bg(O) (p, r) as a compact subset and if

IRm (x, t)I ~ K for all x EU and t E [O, T],