1547671870-The_Ricci_Flow__Chow

4. ESTIMATING INJECTIVITY RADIUS IN POSITIVE CURVATURE 313

Note that a proper j-gon may be regarded as a proper k-gon for any
k > j merely by choosing extra vertices. In fact, it will be easier to deal
with limits if we work with a more general collection of objects.

DEFINITION B.67. A (nondegenerate) geodesic k-gon is a (nonde-

generate) proper geodesic j -gon for some j = 1, ... , k.

If the theorem is false, there is a nondegenerate geodesic 2-gon r in Mn
of total length 2>. < 7r / VK. Let 0 E Mn be any choice of origin, and
let Cr denote the sub level sets of the Busemann function based at 0. By

Corollary B.61, the collection {Cr: 0 < r < oo} exhausts Mn. Thus there

is s E (0, oo) such that r c Cs. By Proposition B.62, Cs is compact. So

define

A ~ { nondegenerate geodesic 2-gons in Cs of total length ::::; 2>.}

and

2 ~ {geodesic 2-gons in Cs of total length ::::; 2>.}.

Let sn-^1 cs denote the unit sphere bundle of Mn restricted to Cs. Observe

that

A<;;;; 2 <;;;; (sn-^1 cs x [O, 2>.l) x (sn-^1 cs x [O, 2>.l),

because any r E 2 is described uniquely by data (p, V,.e, p', V',R'), where

/'1 is the geodesic path determined by /'1 (0) = p, 11 (0) = V E TpMn, and

L (!'1) = .e E [O, 2>.], while ')'2 is the geodesic path determined by ')'2 (0) = p',

12 (0) = V' E Tp'Mn, and L (!'2) = R' E [O, 2>.]. (V may be chosen arbitrarily

if .e = 0, likewise for V' and R'.) It is easy to see that 2 is closed in the induced

topology, hence is compact.

LEMMA B.68. Let (Mn,g) be a complete noncompact manifold with sec-

tional curvatures bounded above by K. Let A <;;;; 2 be defined as above. Then

A is closed, hence is compact.

PROOF. Suppose {ai} is a sequence from A such that °'i -> 0: 00 E 2.

Equivalently,

(Pi, Vi, .ei, p~, V{, RD -> (Poo, Voo,Roo, P'oo, V~, R'oo).

We first claim 0: 00 is not a degenerate proper 1-gon. This can happen

only if R 00 = R'oo = 0, hence only if p 00 = P'oo· But since Cs is compact, we

see that this is impossible, because

O < inj (Cs)::::; inj (Pi)= d(pi,cut (Pi))::::; max {.ei,.ea ""'0

as i-> oo.

We next claim that 0: 00 is not a degenerate proper 2-gon. This can

happen only if the path {3 from p 00 to P'oo # p 00 is the path {3' from P'oo

to p 00 , traced in the opposite direction. Then for i sufficiently large, there

are distinct but nearby geodesics !Ji from Pi to p~ and {3~ from p~ to Pi,

such that max { L (!Ji) , L (!JD} ::::; ~7r / VK. But this is impossible, because