466 A. BASIC RICCI FLOW THEORY• one says (M, g (t)) forms a Type IV singularity as t-> 0 if
sup t !Rm(·, t)I < oo.
Mx(O,T]
We have the following examples of singularities. A neckpinch forms a
Type I singularity (Section 5 in Chapter 2 of Volume One). A degenerate
neckpinch, if it exists, forms a Type Ila singularity (Section 6 in Chapter 2
of Volume One). Many homogeneous solutions form Type III singularities
(Chapter 1 of Volume One).
CONJECTURE A.37 (Degenerate neckpinch existence). There exist so-
lutions to the Ricci flow on closed manifolds which form degenerate neck-
pinches.
The analogue of the above conjecture has been proved for the mean
curvature flow [9].CONJECTURE A.38 (Nonexistence of Type Ilb on closed 3-manifolds). If
(M^3 , g (t)) , t E [O, oo ), is a solution to the Ricci flow on a closed 3-manifold,
then g (t) forms a Type III singularity.
Similar to the division of types for finite time singular solutions, we may
divide ancient solutions into types.
DEFINITION A.39 (Ancient solution types). Let (Mn, g (t)) be a solution
of the Ricci flow defined on (-oo, 0).
•We say (M,g (t)) is a Type I ancient solution ifsup !ti !Rm(·, t)I < oo.
Mx(-oo,-1]• We say ( M, g ( t)) is a Type II ancient solution if
sup !ti !Rm(·, t)I = oo.
Mx(-oo,-1]
3.2. Ancient solutions have nonnegative curvature. Every an-
cient solution (of any dimension) has nonnegative scalar curvature. (See
Lemma 9.15 on p. 271 of Volume One.)
LEMMA A.40 (Ancient solutions have R 2: 0). Let (Mn,g(t)) be a com-
plete ancient solution of the Ricci flow. Assume that the function Rmin (t) -:-
infxEMn R (x, t) is finite for all t :S 0 and that there is a continuous func-
tion K (t) such that !sect [g (t)JI :SK (t). Then g (t) has nonnegative scalar
curvature for as long as it exists.
A particular consequence of the Hamilton-Ivey estimate is that ancient
3-dimensional solutions of the Ricci flow have nonnegative sectional curva-
ture. (See Corollary 9.8 on p. 261 of Volume One.)
COROLLARY A.41 (Ancient 3-dimensional solutions have Rm 2: 0). Let(M^3 , g (t)) be a complete ancient solution of the Ricci flow. Assume that
there exists a continuous function K (t) such that !sect [g (t)]I :SK (t). Then
g (t) has nonnegative sectional curvature for as long as it exists.