1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. SPATIAL POINT PICKING METHODS 51

(3) d (xi, 0) -t oo,

(4) the balls B (xi, ri) are disjoint.

PROOF. (1) Since supM R = oo, there exists a sequence of points {Yi}

such that R (yi) -too as i -too. Let Xi EB (Yi, 2) be a point such that

(18.21) R (xi)· d^2 (xi, 8B (Yi, 2)) = max R (x) · d^2 (x, 8B (Yi, 2)).

xEB(yi,2)

Let ri ~ ~d (xi, 8B (Yi, 2)); clearly ri E (0, l]. For any x E B (xi, ri) c

B (Yi, 2) we have d (x, 8B (Yi, 2)) 2: ri, so that

R (x) rt SR (x) · d^2 (x, 8B (Yi, 2)) SR (xi) (2ri)^2.

That is,

R (x) S 4R (xi) for all x EB (xi, ri).

(2) By (18.21) we have

4R (Yi)= R (Yi)· d^2 (yi, 8B (Yi, 2)) SR (xi) (2ri)^2.

Hence

R (xi) rf 2: R (Yi) -too.

(3) Since R (Yi) -t oo implies that d (Yi, 0) -t oo as i -t oo, by the

triangle inequality we have d (xi, 0) 2: d (Yi, 0) - 2 -t oo as i -t oo.

(4) Since Ti s 1 and d (xi, 0) -t oo, by passing to a subsequence, we
may assume that the balls B (xi, ri) are disjoint. 0
A modification of the above result yields the following improvement.
PROPOSITION 18.13 (Point picking on noncompact manifolds with un-
bounded curvature, version II). Let { C:i} ~ 1 be a bounded sequence of positive
real numbers.^6 If (Mn,g,O) is a complete noncompact pointed Riemann-
ian manifold with sup M R = oo, then there exist sequences {Xi} ~ 1 and
Ti E (0, 1] such that

(1) SUPB(xi,ri) RS (1 + C:i) R (xi),

(2) R (xi) rr -too,

(3) d (xi, 0) -t oo,

(4) the balls B (xi,ri) are disjoint.

REMARK 18.14. Remark 18.11 equally applies to this proposition.

PROOF. (1) Since supM R = oo, we may choose a sequence of points
Yi E M such that

(18.22) R (Yi) c:f -t oo.

(Since {c:i} is bounded, we also have R(yi) -too.) Define Xi EB (yi, 1) so
that

(18.23)

(^6) In applications one often chooses {c-i} so that Ei---+ 0 as i---+ oo.