82 19. GEOMETRIC PROPERTIES OF K-SOLUTIONS
(ii) There is a constant C < oo such that the scalar curvature R9 (x, t) ~
C for all (x, t) EM x (-oo, O].
The relevance of 11;-solutions to singularity analysis is as follows. A fi-
nite time singularity model satisfies all of the properties in the definition
of a K-solution except possibly the nonnegativity of the curvature operator.
However in dimension n = 3, by applying the Hamilton-Ivey 3-dimensional
curvature estimate to Theorem 19.4, finite time singularity models have
nonnegative curvature operator.
COROLLARY 19.8 (3-dimensional finite time singularity models). When
n = 3, under the same hypotheses of Theorem 19.4, there exists a subse-
quence such that (M^3 ,gi(t),pi) converges to a complete K-solution
(M~, 9oo(t),Poo)in the C^00 Cheeger-Gromov sense, where K > 0 is a constant depending only
on g (0) and T.
PROBLEM 19.9 (Rotationally symmetric ancient solutions). Does there
exist a rotationally symmetric ancient solution with bounded nonnegative
curvature operator in dimension at least 3 which is not K-noncollapsed at
all scales for any K > O?
1.2. Some examples of 11;-solutions.
For the first class of examples of K-solutions, we have the products of
spheres and Euclidean spaces and the quotients of these products.
EXAMPLE 19.10 (Cylinders and their quotients). Consider the compact
and noncom pact (smooth manifold) quotients ( sn-k x JRk) /r of the shrink-ing round cylinder with n - k :'.'.". 2 and k :'.'.". 0. Here, the largest K > 0 where
the ancient solution is 11;-noncollapsed on all scales depends on the number
Ir! of elements of rand the action of r on JRk.^2 1n particular (n = 3 and
k = 0) for smooth quotients S^3 /r, where r c SO ( 4), there does not exist a
uniform lower bound for K. (This is in general the case for odd-dimensional
spherical space forms; on the other hand, the only even-dimensional spheri-
cal space forms are sn and ]Rpn.)
The above examples are also useful to keep in mind while consideringthe noncompactness and dimension assumptions on the manifold Mn in the
hypotheses of the '11;-gap theorem' of §6 of this chapter.
REMARK 19.11. The cigar soliton and the King-Rosenau solution (a.k.a.
sausage model) are not K-solutions because they violate the condition of K-
noncollapsing at all scales. In fact, Hamilton proved in [92] that the round
shrinking 2-sphere and its Z2-quotient (i.e., the constant curvature IRP^2 )
are the only 2-dimensional K-solutions. See also §1 of Chapter 9 (especially
Corollary 9.19) in [45].
(^2) For example, if k = 0, then the largest K, is at least en/ II'I, where en > 0 depends
only on n.