474 A. BASIC RICCI FLOW THEORY
PROPOSITION A.66 (n 2:: 2 backward limit of Type II ancient solutionwith Rm 2:: 0 and sect> 0). Let (Mn,g(t)), t E (-oo,w), w E (O,oo], be a
complete Type II ancient solution of the Ricci flow with bounded nonnegative
curvature operator and positive sectional curvature. Assume either
(1) M is noncompact,
(2) n is even and M is orientable, or
(3) g (t) is K,-noncollapsed on all scales.Then there exists a sequence of points and times (xi, ti) with ti __, -oo such
that (M, 9i (t), Xi), where gi(t) ~Rig (ti+ Ri^1 t), limits in the 000 pointed
Cheeger-Gromov sense to a complete nonflat steady gradient Ricci soliton
(M~, g 00 (t), x 00 ) with bounded nonnegative curvature operator.Theorem 9.30 on p. 344 of [111]:THEOREM A.67 (Ancient has AVR = 0). Let (Mn, g(t)), t E (-oo, O], be
a complete noncompact nonflat ancient solution of the Ricci flow. Suppose
g(t) has nonnegative curvature operator and
sup I Rm g(t) (x) lg(t) < oo.
(x,t)EM x (-oo,O]Then the asymptotic volume ratio AVR(g(t)) = 0 for all t.
Theorem 9.32 on p. 345 of [111]:THEOREM A.68 (Type I ancient has ASCR = oo). If (Mn,g(t)), -oo <
t < w, is a complete noncompact Type I ancient solution of the Ricci flow
with bounded positive curvature operator, then the asymptotic scalar curva-
ture ratio ASCR(g(t)) = oo for all t.5. Classical singularity theory
In this section we continue the discussion of Hamilton's singularity the-
ory and recall some further results concerning the classifications of singu-
larities, especially in dimension 3. An exposition of some of these results,
which were originally proved by Hamilton in [186] and [190], is given in
[111]. One of the differences between Hamilton's and Perelman's singular-
ity theories is that in Hamilton's theory, singularities are divided into types,
e.g., for finite time singularities, Type I and Type Ila. In Perelman's the-
ory, a more natural space-time approach is taken where singularity analysis
is approached via the reduced distance function. Throughout most of this
section we shall consider the case of dimension 3.
We first consider the case of Type I singularities, which was essentially
treated in Volume One (see Theorems A.51 and A.52 above). Applying
the compactness theorem and the classification of Type I ancient surface
solutions to Theorem A.51 yields the following.