134 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONSthere exist M < oo and r E [1, ~) such that
(30.l)on M x (a, w).M
JRmJ (x , t) :=:; ( w-t r
Note that the time interval (a,w) is finite and the Type A condition is essen-
tially for the solution near time w. We could impose the Type A condition for anancient solution by requiring JRmJ (x , t) :=:; 1 ~r on M x (-oo, 0). However, if t = 0
is the singular time, then the conditions near times 0 and -oo are competing in
the sense that increasing r makes the condition weaker near 0 but stronger near-oo. So, except for the r = 1 (i.e., Type I) case, one usually imposes the Type A
condition either near the singular time or near time -oo.
We observe that the Type A condition on the curvature leads to corresponding
bounds for the derivatives of curvature. In general, suppose that (Mn, g (t)), t E
(a, w), is a complete solution to the Ricci fl.ow. Fix any rJ E (0 , w2°'] and let
l E [a + rJ, w). Recall that Shi's local derivative estimates imply that if JRmJ :=:; Kon M x (a, l], where K :'.'.'. 1, then for a ny k, f, EN U {O}, there exists Cn,k,e,ry < oo
(independent of land K) such that
(30.2) I :tkk \le Rml (x, l) :=:; Cn,k,e,ryK^1 +k+!
for all x E M (this follows from applying (20.38) in Part III to the solution with atime lag 'T)K-^1 :=:; 'T); i.e., choose the initial time for Shi's estimates to b e t-'T)K-^1 ).
Now suppose that the solution g (t) is Type A. Since (30.1) implies thatM
JRmJ (x , t) :=:; ( w-t rfor (x, t) EM x (a, l], we conclude from (30.2) with K = (w~f that we have the
following derivatives of curvature estimates.LEMMA 30.2 (Shi's estimates for Type A solutions). If (Mn, g (t)), t E (a,w), is
a complete Type A solution of the Ricci flow, then for any k , f, :'.'.'. 0 and rJ E (0 , w2°']there exists Cn,k,e,ry,M < oo such that
(30.3) -\le Rm (x l) < n,k,e,ry,M
I
[)k I c
8tk ' - (w-t)(l+k+!)r
on M x [a+ rJ,w). In particular, taking k = 0 and f, = 1, we obtain
(30.4)
onM x [a +rJ,w).
J'V RI (x,l) :=:; Cn J\7 RmJ (x , l) :=:; Cn,ry,~
(w - [)2rThe reader may wish to compare this with the modified Shi's derivative esti-
mates in Chapter 14 of Part IL