56 18. GEOMETRIC TOOLS AND POINT PICKING METHODS
monotonicity principle, i.e., Lemma G.38,^9 then implies that the Euclidean
comparison angle
which in turn implies that
di(Pi, qi)-+ di(Pi, Xi)+ di(qi, Xi)= 2p,where di denotes the distance with respect to the metric 9i ( 0), so that
doo(Poo, qoo) = doo(Poo, Xoo) + doo(qoo, Xoo) = 2p
in M 00 , where p 00 and q 00 are the limits of Pi and qi. Since p > 0 is arbitrary,
this implies that (M 00 , 900 (0)) contains a line passing through Xoo, namely
the concatenated path -1 00 U a 00 •
It follows, as in the proof of Theorem 18.17, that (M~, 900 (t)) splits as
the product of an (n - 1)-dimensional nonfl.at solution with bounded Rm 2:
0 and a line. D
N ONEXAMPLE. Consider the rotationally symmetric expanding soliton
on ]Rn, n 2: 3, with positive sectional curvature; we discussed its construction
in §5 of Chapter 1 in Part I. The sectional curvatures of these solutions decay
quadratically (Proposition 1.43 in Part I) and the warping function grows
linearly (Proposition 1.42 in Part I). In this case one does not have dimension
reduction; indeed a blow-down Hmit about a sequence of points tending to
spatial infinity will limit to a rotationally symmetric cone.
2.4.2. Existence of a line in a limit of a K-solution with ASCR = oo.
By Theorem 18.10 and Theorem 18.19 we have the following.COROLLARY 18.21 (Dimension reduction for ancient K-solutions). If
(Mn, 9 ( t)), t E ( -oo, w), where w > 0, is a noncom pact K-solution with
ASCR(t) = oo, then (M, 9 (t)) dimension reduces to the product of an
( n - 1 )-dimensional K-solution with bounded Rm 2: 0 and a line.
REMARK 18.22. As we shall see below, by Theorem 20.1, a noncompact
K-solution with Harnack must have ASCR( t) = oo for all t E ( -oo, w).2.4.3. Alternate method for dimension reduction - curvature bumps.
Another way to obtain dimension reduction is via the following result
of Hamilton (see §21 and §22 of [92]), which also relies on the Toponogov
comparison theorem (see also Theorem 8.51 in [45]).
THEOREM 18.23 (Finite number of curvature bumps in manifolds withsect 2: 0). Given any c > 0 and n 2: 2, there exists .A < oo depending only
on c and n such that if (Mn,9) is a complete Riemannian manifold with
(^9) Let (Mn, g) be a complete Riemannian manifold with nonnegative sectional curva-
ture. Given p EM and two minimal geodesics a(s) and /3(t) such that a(O) = /3(0) = p,
the Toponogov monotonicity principle says that the Euclidean comparison angle fJ(s, t) ~
2a ( s) p/3 ( t) is nonincreasing in both s and t.