- SPATIAL POINT PICKING METHODS 49
since f 13 Re ({3' ( s) , {3' ( s)) ds :::; ( n - 1) KL (/3).
REMARK 18.9. The above results, which imply bounds for the time de-
rivative of the distance function under the Ricci fl.ow, are useful in the study
of ancient solutions. For discussions of this use, see for example Lemma
8.33, Lemma 9.51, and Proposition 9.81, all in [45].2. Spatial point picking methods
In this section we discuss spatial point picking methods primarily due
to Perelman. Earlier, basic examples of the use of point picking to obtain
estimates occurred in Hamilton's work on singularity analysis (see especially
[92] and [94]) and Schoen's work [166] and [167] on constant scalar curva-
ture metrics in a conformal class (related to the Yamabe problem).
Let ASCR denote the asymptotic scalar curvature ratio (see p. 472 in
Part I or definition (19.8) below). In this section we consider point picking
in the following scenarios:
(1)
(2)(3)(4)a Riemannian manifold with sup R < oo and ASCR = oo;
a Riemannian manifold with sup R = oo (which implies ASCR =
oo);
a sequence of Riemannian manifolds where the change in R is un-
bounded within some finite distance;
dimension reduction, where one seeks a limit which splits off a line.2.1. Point picking when sup R < oo and ASCR = oo.
The asymptotic scalar curvature ratio is useful in studying the geometry
at infinity of noncom pact ancient solutions.^3 One reason for why this is true
is the following result (Lemma 22.2 of Hamilton's [92]; see also Theorem
8.44 in [45]).^4
THEOREM 18.10 (Point picking on complete noncompact manifolds with
ASCR = oo). If (Mn, g, 0) is a complete noncom pact pointed Riemannian
manifold with^5
sup R < oo and ASCR (g) = oo,
M(^3) Roughly speaking, ASCR = oo says that the curvature decays slower than quadrat-
ically. In particular, a manifold asymptotic to a cone has ASCR < oo.
(^4) In the statement of Theorem 8.44 in [45] the assumption that supM R < oo was
inadvertently omitted.
(^5) In this theorem, as well as the point picking results below, we do not need to assume
that (Mn, g) has Re;::: O; note that, in definition (19.8) of ASCR, we only need to assume
that (Mn,g) is a complete noncompact Riemannian manifold.