1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

4. DERIVATIVE ESTIMATES AND SOME CONJECTURES 151

for x E M and t :::; 0. More generally, for any R and m there exists 'Tl ( R, m) <

oo (independent of r;,) such that

I ::c \7mRm[ (x, t) :::; 17 (R, m) R (x, t)^1 H+~

for any x EM and t:::; 0.

We may call these universal scaled derivative bounds since the quantities

R (x, t)-(l+c+~) I gt~ vm Rm[ (x, t) are scale-invariant.

PROOF. Using the improved compactness result of Corollary 20.ll(ii),

we can prove Corollary 20.19 above for any (M^3 ' g (t)) E u-;;>0 9.Jt~~~h just

as we proved the derivative estimate of Theorem 20.17. This is possible

because u-;;>0 9J13,i,; equals the union of ui-;;>0 9.Jt~~~h with the collection of

spherical space form solutions and because the estimates clearly hold for the

spherical space form solutions with T/o = 1.^21 Hence the derivative estimates

hold for the collection LJi-;;>O 9J13,i-;;· D

REMARK 20.20 (Derivatives estimate fails on the cigar soliton). Recall
that the cigar soliton is :E ~ JR^2 with the metric 9IJ = ds^2 + tanh^2 s dB^2
and scalar curvature RIJ = 4 sech^2 s. From this it is easy to see that
the first space derivatives estimate does not hold on the cigar soliton, i.e.,
supIJ R-^312 JV' RI = oo. Moreover, dilating the cigar by curvature about a
sequence of points tending to infinity, one obtains as the limit a half-line
with basepoint equal to its vertex.

4.2. Some applications of the derivative estimates.

Let (Mn, g (t)), t E (-oo, O], be a noncompact r;,-solution with Harnack;

note that R > 0. Fix the time and a point p EM. Given any x EM, let

a : [O, s] --+ M be a minimal unit speed geodesic joining p to x. By Theorem

20.17, we have

:

8

R-^1 l^2 (a(s),t) = \v (R-^1!^2 ) ,a'(s)):::; ~R-^312 JVRJ:::; ~·

Integrating this along a implies

(20.51) ( 1/2 'Tl )-

2

R (x, t) 2::: R- (p, t) + 2dg(t) (x,p).

In particular, we obtain

COROLLARY 20.21. For a noncompact r;,-solution with Harnack

4

liminf R (x, t) dg(t) (x,p)^2 2::: 2.

~W~~~oo 'Tl

(^21) Note that the shrinking round 3-sphere g(t) = (1-4t)g 8 s(i) has R(x,t) = 1 ~ 4 t
and l'Vm Rml = 0 form;:::: 1.