1547671870-The_Ricci_Flow__Chow

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then

7. DERIVATIVE ESTIMATES

IVm Rm (x, t)ig(x,t) :S ~:/~

a

for all x E Mn and t E (0, K].

Note that the estimates in Theorem 7.1 follow the natural parabolic
scaling in which time behaves like distance squared. Note too that we have
stated the estimates in a form that deteriorates as t ~ 0. This is the best one
can do without making further assumptions on the initial metric. Indeed,
it is easy to construct examples of rotationally symmetric metrics on sn
with !RI :S 1 and IV RI arbitrarily large by writing the metric as a warped
product. (Recall that we used warped product metrics on spheres in Section
5 of Chapter 2.)
Theorem 7.1 has several important consequences, of which the following
two are particularly useful.

COROLLARY 7.2 (Long-time existence). If go is a smooth metric on a

compact manifold Mn, the unique solution g (t) of the Ricci flow such that

g (0) = go exists on a maximal time interval 0 :S t < T ::::; oo. Moreover,

T < oo only if

lim ( sup !Rm (x, t)i) = oo.

t/T xEMn

This result equivalent to Theorem 6.45.

COROLLARY 7.3 (Uniform bounds for sequences). Let {(Mi, gf) : i E N}

be a sequence of compact Riemannian manifolds. If their curvatures are
uniformly bounded in the sense that

IRm [gf] 19? :SK

for some constant K < oo independent of i E N, then there exists a constant

c > 0 depending only on n such that for each i E N, the unique solution

gi (t) to the Ricci flow such that gi (0) = gf exists on Mi for the uniform
time interval t E [O, c/ K]. Moreover, there is for each m E N a constant
Cm < oo depending only on m and n (that is, independent of i E NJ such
that

sup IVmRm[gi(t)]l 9 i(t):S (~r·

xEMr t

In particular, for any time to E (0, c/ K), there is for each m E N some
c:n = c:n (m, n, K, to) such that

sup IVmRm[gi]l 9 i :S c:n.
Mnx[to,c/K]
This fact is an immediate consequence of Theorem 7.1 and the minimal
existence time result (Corollary 7.7) we shall prove below. An important
application of Corollary 7.3 is to obtain convergence in each cm norm when
one proves the compactness theorem for the Ricci fl.ow, which we state in
Section 3 of this chapter.