l. REDUCED DISTANCE OF TYPE A SOLUTIONS 1351.2. Motivation and set-up for the reduced distance based at the
singular time.
Let (Mn,g(t)), t E (a,w), where - oo <a< w < oo, be a complete singular
solution of the Ricci flow. Recall from Chapter 7 in Part I that for any p 0 E Mand to E (a, w), the reduced distance based at (p 0 , t 0 )
fp 0 ,t 0 : M x (a,to)--+ JR
is defined by£"'·'" ( q, t) ~ ~inf J:" v'l0=1 ( R (> ( t) , t) + 11 ( t) l!uJ) dt,
2 to - t ' t(30.5)
where the infimum is taken over 'Y: [f, to] --+ M with 'Y(f) = q and 'Y (to) =Po·
Given a sequence (pi, ti) EM x (a, w) with ti/' w, we may ask if there exists
a subsequence such that the reduced distance functions fp,,t, converge t o a limit
function defined on M x (a, w).
DEFINITION 30.3. If such a limit exists, we shall call it a reduced distancebased at the singular time.
We first consider a general setting which includes the case for which we can
prove existence; see Theorem 30. 10 below. Let (Mf,gi (t) , pi ), t E [ai,wi], Pi E
Mi, be a sequence of complete pointed solutions to the Ricci flow with wi /' w 00
and ai ::::; a 00 ~ limi--too ai· Suppose that this sequen ce converges in the C^00
pointed Cheeger- Gromov sense to a complete limit solution (M~,g 00 (t) , p 00 ) , t E
[a 00 , w 00 ). That is, suppose that there exist smooth embeddings
(30.6)where {Ui}iEN is an exhaustion of M 00 by open sets with compact closure, such
that
gi (t) ~ :gi(t)--+ 9oo(t)
in Ck(JC), for each k EN and for each compact subset JC of M 00 x [a 00 , w 00 ).
Furthermore, we sh all assume that each solution gi (t) h as bounded curvature
on Mi x [ai, wi], where the bound may depend on i. Let the functions
on Mix [ai,wi) be the reduced distance and t he L-distance for gi (t) based at
(Pi, wi), respectively (see Chapter 7 of Part I for the definition of the L-distance).
Define the corresponding "transplanted" functions ii and Li on ui x [ai, Wi) by
(30. 7)We shall estimate ii and Li on ui x [aoo, wi) c Moo x [aoo,woo) and we shall prove
their convergence under the Type A curvature assumption.
1.3. Pointwise bounds for the reduced distances.
By a rather direct method we first establish a p ointwise bound for Li under the
Type A assumption, which is uniform (i.e., independent of i) on compact subsets of
M 00 x [a 00 , w 00 ). This, along with the derivative estimates in t h e next subsection,
enables us to take limits of the reduced distance functions. In the next section
we shall obtain sharper bounds in the Type I case (see Lemma 30. 15 below). This