- ASCR AND AVR OF 11;-SOLUTIONS 127
Case B. 0 < ASCR (to) < oo for some to ::::; 0. In this case a semi-global
blow-down limit is a piece of the nonfiat asymptotic cone which contradicts
the strong maximum principle implying that it is fiat.
By the definition of ASCR, there exists a sequence of points Xi E M
such that as i-+ oo,d 9 (to)(xi,P)-+ oo, R(xi, to) d;(to)(xi,P)-+ ASCR(to).Define the rescaled solutions {(Mn,gi(t))}iEN as in (20.1). Note that this is
a blow-down sequence since R(xi, to) -+ 0 as i-+ oo. We have as i -+ oo,(20.2) d 9 i(o)(xi,P) = R(xi, to)^112 dg(to)(Xi,P)-+ v ASCR(to) E (0, oo).
Let band B be any two constants with 0 < b < .J ASCR(to) < B < oo
and letNi(b, B) ~ B 9 i(o)(P, B) \ B 9 i(o)(P, b) c M
be the annulus, with respect to gi (0), with inner radius b and outer radius
B. Then by (20.2) we have Xi E Ni(b, B) for i sufficiently large.
By the trace Harnack estimate, which implies that gtRYi ;::: 0, and since
ASCR(to) E (0, oo), we have the curvature bound
R .( t) < R .(x O) < 2ASCR(to) < 200ASCR(t^0 )
9i x, - 9i ' - d2 9i(O) ( x,p ) - 81b2
for all x E Ni ( ig,^1 i~) and t E ( -oo, O], provided i is sufficiently large.
Again (as in Case A), the assumption that g(t) is f£-noncollapsed at
all scales implies that inj 9 i(o) (xi) 2: 8 for some 8 > 0. Applying the (lo-
cal) compactness theorem, i.e., Theorem 3.16 in Part I, to the sequence
of (incomplete) solutions {(Ni ( ig,^1 i~), gi ( t), Xi) LEN' we obtain that for a
subsequence,
(Ni (b, B), gi (t), Xi) -+ (Noo (b, B), goo (t), Xoo)as i -+ oo. Here (N 00 (b, B) , g 00 ( t)) is a nonfl.at (incomplete) solution to the
Ricci fl.ow with bounded nonnegative curvature operator.
Since gi (0) = R(xi, to)g(to), since R(xi, to) -+ 0 as i-+ oo, and since the
sectional curvatures of gi (0) are nonnegative, by Theorem I.26 in Appendix
I we have that
(Mn,gi (0) ,p)-+ (CW,doo,Poo)converges in the pointed Gromov-Hausdorff topology as i -+ oo to a unique
limit, where
CW~ (llho x W)/(O x W)is a Euclidean metric cone over a metric space (W, dw). The limit metric
space (CW, d 00 ,p 00 ) is called the Gromov-Hausdorff asymptotic cone^5
of ( M, g (to)).
(^5) See §1 of Appendix H.