1547671870-The_Ricci_Flow__Chow

7. HIGHER DERIVATIVE ESTIMATES AND LONG-TIME EXISTENCE 201

THEOREM 6.45. If go is a smooth metric on a compact manifold Mn,

the unnormalized Ricci flow with g (0) =go has a unique solution g (t) on a

maximal time interval 0 :::; t < T :::; oo. Moreover, if T < oo, then

lim ( sup !Rm (x, t)i) = oo.
t/T xEMn
We shall in fact prove the contrapositive of the theorem. Namely, we
will show that if the maximum curvature were to remain bounded along
a sequence of times approaching T, the solution could be extended past
T. In order to make this argument rigorous, we must first discuss a priori
estimates for any solution whose curvatures remain suitably bounded.

7.1. Higher derivative estimates. In Section 5 of Chapter 3, we saw

how the Ricci flow may be regarded heuristically as a nonlinear heat equation

for a Riemannian metric. This viewpoint admits a rigorous interpretation

in the sense that the curvatures of an evolving metric all satisfy parabolic

equations. In particular, bounds on the curvature of an initial metric go

automatically induce a priori bounds on all derivatives of the curvature for

a short time. We call these Bernstein-Banda-Shi (BBS) derivative

estimates, because they follow the strategy introduced by Bernstein [16,

17, 18 ] for proving gradient bounds via the maximum principle, and were

derived for the Ricci fl.ow in [9] and [117, 118]. (See also [11] and Section

7 of [63].)

Because the BBS estimates for the Ricci fl.ow are not surprising in that

they follow the natural parabolic scaling in which time scales like distance

squared, we shall postpone our technical discussion of them until Chapter 7.

(Certain local analogues of the BBS estimates will also be discussed in the

planned successor to this volume.) It will be enough for our present needs

to assume the following result from Chapter 7.

THEOREM 6.46 (Theorem 7.1). Let (Mn,g (t)) be a solution of the Ricci

flow for which the maximum principle holds. {This is true for instance if

Mn is compact.) Then for each a > 0 and every m E N, there exists a

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

then

a

!Rm (x, t)l 9 :::; K for all x E Mn and t E [O, K],

a

for all x E Mn and t E (0, KJ.

As stated above, the BBS derivative estimates deteriorate as t ~ 0. This

is due to the fact that bounds on the Riemann tensor of an initial metric

imply nothing about its derivatives. Furthermore, the BBS estimates above

allow the factors C 7 to grow as the time interval over which we estimate

becomes larger. Because our ultimate goal is to establish long-time existence

of the fl.ow, we want a practical criterion that will let us know when a solution

which exists up to an arbitrarily large time T can be extended past T. We