146 20. COMPACTNESS OF THE SPACE OF /\;-SOLUTIONSfor any y E Bgk(O) (xk, r). This estimate and the 11;-noncollapsing condition
enable us to apply Hamilton's compactness theorem to get a convergent sub-sequence (Mk, 9k (t), Xk) --+ (M~, g 00 (t), x 00 ). Clearly the limit is a (com-
plete) 11;-solution with Harnack. Therefore Theorem 20.9 is proved modulo
the proof of (20.42).
We now give the details of the proof of (20.42). We prove the follow-
ing lemma regarding uniform bounded curvature at points with uniformly
bounded distance from the origin for a sequence of solutions of the Ricci
fl.ow. This result is also known by the catchphrase 'bounded curvature at
bounded distance'. The idea of the proof is similar to the argument in the
second half of the proof of the claim stated at the beginning of Step 2 above.LEMMA 20.15. Let {(Mk,9k (t) ,xk), t E (-ak,O]}kEN' be a sequence of
complete solutions of the Ricci flow with nonnegative curvature operator andwith gtRgk (x, t) 2:: 0 for any (x, t) E Mk x (-ak, 0). Suppose that
R 9 k (xk, 0) = 1 and Vol 9 k(o) B 9 k(o) (xk, ro) 2:: vo
for some constants ro > 0 and vo > 0 independent of k. Then, for any
r > 0, either
(i) there exists a constant Co < oo independent of k such that
ak sup R 9 k (x, 0) :S Co
xEBgk(o)(xk,r)
or(ii) there exists a constant C1 < oo independent of k such that
sup R 9 k (x, 0) :S C1.
xEB 9 k(o)(xk,r)
PROOF. We will prove the lemma by contradiction. If the lemma isfalse, then there exist r * > 0 and a sequence of complete solutions
{(Mk,9k (t) ,xk), t E (-ak,O]}
which satisfies the following conditions:
(1) The curvature operator is nonnegative, i.e., Rm 9 k ;=:: 0, and thescalar curvature is pointwise nondecreasing, i.e., gtR 9 k (x, t) > 0 for all
(x, t) E Mk x (-ak, 0).
(2) There exist constants ro > 0 and v 0 > 0 such that
R 9 k (xk, 0) = 1 and Vol 9 k(o) B 9 k(o) (xk, ro) ;::: vo.
We have
(3)
ak sup R 9 k (x, 0) --+ oo
xEB 9 k(o)(xk,r)
and
(4)
sup R 9 k (x, 0) --+ oo.
xEBgk(O) (xk,r)