64 18. GEOMETRIC TOOLS AND POINT PICKING METHODS
4.2. The diameter of a level set of a Busemann function in an
c:-neck.
The following lemma will be used to prove Proposition 18.33. Given asubset SC M, let diam (S) ~ SUPx,yES d (x, y).
LEMMA 18.30 (Bounds for the diameter of a level set of a Busemann
function intersecting the center sphere of a neck). For any n 2: 3 there
exists c(n) > 0 which has the following property. Suppose (Nn, h) is a
complete noncompact Riemannian manifold with positive sectional curvature
and suppose
'l/J: sn-^1 x [-c:(n)-^1 +1, c:(n)-^1 - 1] ~ 91 c N
is an embedded c:( n )-neck. Let / : [O, oo) ~ N be a (unit speed) ray and let
b 1 : N ~IR be the Busemann function defined by /.^14 Let qo ~ 'ljJ (zo, 0) E
'ljJ ( sn-l x { 0}) be a point on the center sphere S and let bo ~ b 1 ( qo). Then
the level set b:;1 (bo) has the property^15(18.51) *JrcnR(qo)-^112 :S diam (b~^1 (bo)) :S llKcnR(qo)-^1!^2 ,where Cn ~ ,,/(n - 1) (n - 2).PROOF. Let c: (n) E (0, 1), to be chosen later sufficiently small. ByLemma 18.28, the manifold minus the center sphere of the neck, i.e., N -
'ljJ ( sn-l x {O}), is disconnected and consists of two connected components:
one component, which we call Ninn, is diffeomorphic to IRn and the other
component, which we call Nout, is diffeomorphic to IRn - { o}. Without loss
of generality we may assume
Oinn91 ~ 'l/; (sn-l X {-c:(n)-^1 +1}) C Ninn
and
Oout91 ~ 'l/; (sn-l X { c:(n)-^1 - 1}) C Nout·
Note that Ninn U 91 C N is a closed smooth n-ball. Since / ( s) is a ray, there
exists so E (0, oo) such that for s 2: so we have
1(s) EN-(NinnU91).
Hence for s 2: so, any minimal geodesic joining any point p E 'ljJ ( sn-l x { uo})
with uo :S c:(n)-^1 - 2to1(s) must intersect each slice 'ljJ (sn-l x {u}) for
u E [uo, c:(n)-^1 - 1].
(^14) That is,
b 1 (x) =i= s--roo lim (s - d (I' (s), x))
= sup (s-d('Y(s),x)).
sE[O,oo)
(^15) Keep in mind that diam (sn-l (r)) = 1rr = 7rCnR- (^112).