1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. SPATIAL POINT PICKING METHODS 53

REMARK 18.16. Thus, relative to the curvature at the center, we have

uniformly bounded curvature on balls oflarger and larger radii by (i) and (ii).

This conclusion is essentially the same as in Theorem 18.10 and Proposition

18.13.

PROOF. (i) Define

Fk (w) ~ Rhk (w) (D + 1-dhk (w, Ok))^2

for w E Bhk (Ok, D + 1). Note that Fk = 0 on 8Bhk (Ok, D + 1) and

sup Fk (w) ~ sup Rhk (w)---+ oo

Bhk(Ok,D+l) Bhk(Ok,D)

as k ---+ oo. We choose Wk E Bhk (Ok, D + 1) to be a point which satisfies

Fk (wk)= sup Fk (w).

Bhk (Ok,D+l)

Let Bk~ ( 1-~) (D + 1-dhk (wk, Ok)). Then

Rhk (wk) B~ = (~ -V2) Fk (wk)---+ oo

as k ---+ oo. This proves ( i).

(ii) For any w E Bhk (wk, Bk) C Bhk (Oki D + 1) we have by the triangle

inequality

so that

It follows that

Rhk (w) B~ ( )^2 2

2 ::=; Fk(w) ::=; Fk Wk = Rhk (wk) Bk 2

(V2-1). (V2-1)

and hence Rhk (w) ::=; 2Rhk (wk) for all w E Bhk (wk, Bk). This proves the

lemma. 0

2.4. Dimension reduction for ancient solutions.
Dimension reduction is useful in the study of finite time singularity mod-
els. Recall that such solutions are ancient and, when their dimension is 3,
have nonnegative curvature operator. In this subsection we first give a cri-
terion for when an ancient solution on a manifold of a certain dimension
reduces effectively to a solution on a manifold of one less dimension. As
a consequence, we prove a dimension reduction result for /\;-solutions with
ASCR = oo. In the last part of this subsection we mention Hamilton's
'curvature bumps' theorem.