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68 18. GEOMETRIC TOOLS AND POINT PICKING METHODS

4.3. Bounding radii of farther c:-necks by radii of closer E-necks.

We now estimate the scalar curvature of E-necks farther from an origin

(i.e., a fixed point) by the scalar curvature of E-necks closer to the origin.

Note that the radius of an ideal neck, i.e., an exact cylinder, is equal to

Cn/VR.

PROPOSITION 18.33 (In manifolds with sect > 0 the radii of farther c( n )-

necks are almost larger). There exists c:(n) > 0 depending only on n ;::: 3

which has the following property. Suppose (Nn, h) is a complete noncompact

Riemannian manifold with positive sectional curvature. Let 'Y : [O, oo) --+ N

be a (unit speed) ray and let b'Y : N --+ IR be the Busemann function defined

by"(. If for i = 1, 2, mi are two disjoint embedded c:(n)-necks with Yi on the

center sphere Si of the neck ~' then the scalar curvatures of Yi satisfy

(18.59)

provided d (p*, Yl) :S d (p*, Y2), where p* is any point in b::/ (minx EN b'Y ( x)).^21

PROOF OF PROPOSITION 18.33. Let bi ..;... b'Y (Yi) for i = 1, 2. By our

assumptions on the necks and by (18.55) and (18.58), we have

b2 >bl.

Note that b'Y is a convex function (see Proposition B.54 in Volume One).

Therefore we may apply Sharafutdinov's theorem (see Theorem 3 in [171]

or the expository Theorem I.24 in Appendix I) to obtain

diam (b;y^1 (b1)) :S diam (b;y^1 (b2)).

Hence by Lemma 18.30 we have

11 -1/2 -1/2

12

7renR (y1) :S lhenR (y2) ,

that is,

D

5. Localized no local collapsing theorem

The aim of this section is to prove a localized version of the no local
collapsing theorem given by Theorem 6.74 and Theorem 8.26 in Part I. We
shall assume that the reader is familiar with the £-length .C ("!),the reduced
distance .e (q, T) = 2 ~L (q, T) = 4 ~L (q, T), and the reduced volume V (T)

(see Chapter 7 in Part I or §5 in Chapter 19 in this volume for definitions
and properties).

(^21) See Lemma 18.31 for why b'Y attains its minimum.