1547671870-The_Ricci_Flow__Chow

226 7. DERIVATIVE ESTIMATES

then

M (t) ~ 2M (0) for all times 0 ~ t ~min { T , M~O)}.

PROOF. By Lemma 7.4, M (t) is a Lipschitz function of time which

satisfies

dM < CM

3

= C M 2

dt - 2M. 2

in the sense of the lim sup of forward difference quotients, where C depends

only on n. This implies that

1

M (t) ~ 1 c

M(O) - 2t

as long as t E [O, T] satisfies t < 2/ (CM (0)). Let c = 1/C. Then one has

M (t) ~ 2M (0) for all times t satisfying 0 ~ t ~ min { T , c/ M (O)}. D

REMARK 7.6. We encountered a special case of this result in Lemma

5.45.

Combining Corollaries 7.2 and 7.5 yields the following useful observation.

COROLLARY 7.7 (Minimal existence-time). If(Mn,g 0 ) is aRiemannian

manifold such that !Rm [go] 190 ~ K, then the unique solution g (t) of the

Ricci flow with g (0) = go exists at least fort E [O, c/ K], where c > 0 is a

constant depending only on n.

2. Proving the global estimates
   In proving Theorem 7.1, we shall have to estimate the time derivative
   of the quantity IVk Rml^2 on a solution (Mn, g (t)) of the Ricci fl.ow. This is
   the generalization ton dimensions of the computations in Chapter 5 where
   we considered the evolution of IVk Rl

2

on a surface. To prepare for the

calculations at hand, let us consider a simpler but representative problem.

If Q ( t) is a I-parameter family of ( 1, 0 )-tensor fields on a solution (Mn, g ( t))

of the Ricci fl.ow, then formula (6.1) implies that

gtviQj =gt (a~iQj-r7jQk)

=vi (%tQ) j + (viRJ + Y'jR7-\7kRij) Qk.

Hence by Lemma 3.1, we have

at a IVQI^2 = at a ( g ik 91 e viQj v kQe )

= 2\7iQj\7i (%tQ) j + 2\7iQj (viRJ + Y'jR7-\7kRij) Qk

~. ~.

+ 2Ri \7iQ^1 \7kQj + 2RJ \7iQj\7iQe.