1547671870-The_Ricci_Flow__Chow

12. AN INJECTIVITY RADIUS BOUND 151

PROOF. Choose d < njJKmax so that max{L(,Bi)} < d. Let f be
the space of all nondegenerate geodesic 2-gons 'Y in Mn with L ('Yi) < d
for i = 1, 2. Note that r 3 ,8 is a nonempty open 2n-manifold locally
parameterized by points in Mn x Mn. Indeed, by Lemma 5.66 there is
for every pair of points p and q sufficiently close to p and q a unique

nondegenerate geodesic 2-gon ,8* with L (f3i) < d for i = 1, 2. Define

m ~ inf {Lb1) + L('Y2)}.

')'Er

Note that m 2 inj Mn > 0, since the map expxl Bx (0 , p) : Bx (0, p) ---+Mn
is an embedding at any x E Mn whenever p < inj Mn. Let ,8 ( i) E r be a
sequence of nondegenerate geodesic 2-gons with

Hm { L(/3 (i)i) + L(,8 ( i) 2 )} = m.

t-->00

Let p ( i) I-q ( i) denote the endpoints of the geodesic paths ,8 (i)i and f3 ( ih
By compactness of Mn, we may pass to a subsequence such that the limits
Poo ~ limj--. 00 p (j) and q 00 ~ limj--.oo q (j) exist and satisfy dist (p 00 , q 00 ) :::::;
d < 7r / V K max. Hence geodesics

and

exist and satisfy L (,8 ( oo )i) + L (/3 ( oo ) 2 ) = m by continuity.
There are now two cases. If p 00 = q 00 , then some path, say f3 ( oo )i , is

a nondegenerate geodesic 1-gon at p 00 satisfying L (/3 ( oo )i) = m. The loop

,8 (oo)i must be smooth at p 00 , or else the first variation formula (see 1.3 of
[ 27 ]) would let us shorten ,8 ( 00 h inside r.
In the second case, Poo I-qoo. If ,8 ( oo) ~ {,8 ( oo )i, ,8 ( oo ) 2 } is degener-

ate, then f3 ( oo ) 2 must be the same path as ,8 ( oo )i but with the opposite

orientation. By the uniqueness statement in Lemma 5.66, this is possible
only if f3 (i) 1 = ,8 (i) 2 for sufficiently large i, which contradicts the nonde-
generacy of f3 (i). Hence ,8 (oo) is nondegenerate. As above, f3 (oo) must be
smooth, or else the first variation formula would let us shorten ,8 ( oo) inside
r. D

DEFINITION 5.68. We say a geodesic loop 'Y is stable if every nearby

loop 'Y* satisfies L ("!*) 2 L ("!). We say a geodesic loop 'Y is weakly stable

if the second variation of arc length of 'Y is nonnegative.

Clearly, stable implies weakly stable. The following result is well known.

LEMMA 5.69. Let (Mn, g) be a closed Riemannian manifold with sec-

tional curvatures bounded above by Kmax > 0. If a is the shortest geodesic

loop in Mn and L (a)< 2n/vKmax, then a is stable.

PROOF. If not, there exists a loop a* near a with L (a*) < L (a). Pick

distinct points p and q on a that divide a into two geodesic paths 0:1 and

a 2 of equal length. It is always possible to choose distinct points p* and

q on a near p and q respectively that divide a* into paths ai and a2 of