1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

62 18. GEOMETRIC TOOLS AND POINT PICKING METHODS

We can now apply Theorem 18.7(2) to get that for any t E [t$, t#],

:tdh(t) (w,p) ~ -2 (n-1) (~ · 2R(y*, t*)ro + :

0

),

where on the RHS we used the fact that Re :S: R :S: 2R(y, t) (since the

Ricci curvature is nonnegative). Using 0 :S: t# - t$ :S: DR-^1 (y, t), we may

integrate this inequality to obtain

dh(t#) (w,p) - dh(t$) (w,p)

~ -2 (n - 1) (~ · 2R(y, t)ro + 2-) DR-^1 (y, t);

3 ro

it then follows from (18.48) that

dh(t$) (w,p) :S: dh(t) (y,p) + tA^1 l^2 R-^1 l^2 (y, t)

+ 2 (n - 1) (~ro + R-l(y*, t*)) D.

3 ro

Take r 0 = l 0 R-^112 (y, t). We obtain

dh(t$) (w,p) :S: dh(t*) (y*,p) + ~~A^112 R-^112 (y*, t*).

However, this is a contradiction to the definition oft$. This both rules out

Case 2 and completes the proof of Proposition 18.25. D

4. Necks in manifolds with positive sectional curvature

Any singularity model in dimension 3 is either a shrinking spherical space
form 83 /r, a shrinking round cylinder 82 x JR, its quotient under the an-

tipodal map (8^2 x JR) /Z 2 , or noncompact with positive sectional curvature

(see Corollary 17.16). In the latter case, the underlying manifold is diffeo-
morphic to JR^3. In this section we consider s-necks in complete noncompact
manifolds with positive sectional curvature. One may think of ans-neck as
a good region as compared to a good location.
The main result of this section is Proposition 18.33 below, which shall
be used in the first proof of Proposition 19.44 and which is a consequence
of a result of Sharafutdinov [1 71 J; techniques used in this section include
Busemann functions and the Toponogov comparison theorem.

4.1. s-necks.

First we give the definition of ans-neck. Let en~ .J(n - 1) (n - 2) and

let g5n-1, n ~ 3, be the standard metric on the unit sphere 5n-l (so that

the scalar curvature is equal to (n - 1) (n - 2) = c;,).

DEFINITION 18.26 (Embedded s-neck). Let (Nn, h) be a complete Rie-

mannian manifold. Given s > 0 and p E N with R (p) > 0, let r ~

cnR (p)-^112. A geodesic ball B(p, s-^1 r) ~ SJt in (Nn, h) is called an embed-
ded s-neck, if, after scaling h by r-^2 , the metric on B(p,s-^1 r) is s-close