150 5. THE RICCI FLOW ON SURFACES
PROPOSITION 5.65. Let (M^2 ,g(t)) be a solution of the normalized Ricci
flow on a closed surface with Euler characteristic x (M^2 ) > 0, and let
Kmax (t) denote its maximum Gaussian curvature. Then for all 0 :St< oo,
inj (M^2 ,g(t)) 2:: min{inj (M^2 ,go), min J 1f }.
rE[O,t] Kmax (T)We begin with a collection of standard technical tools that are valid in
any dimension. We denote the open ball of radius p centered ~ E TxMn by
Bx(~, p) and the length of a path/ by L (1).
LEMMA 5.66. Let (Mn,g) be a complete Riemannian manifold with sec-
tional curvatures bounded above by Kmax > 0. Let (3 be a geodesic path
joining points p and q in Mn with L ((3) < 1f / .J K max. Then for any points
p and q sufficiently near p and q respectively, there is a unique geodesic
path {3 joining p and q* which is close to (3.
PROOF. By the Rauch comparison theorem (Theorem B.20 of Appendix
B), the mapexpPI Bp (o,1f/JK::): Bp (o,7r/VKmax)----+ Mn
is a local diffeomorphism. We may assume that the geodesic (3 : [O, 1] ----+ Mn
is parameterized so that {J(s) = expp(sV) for some VE Bp(0,7r/)Kmax)
with expP (V) = q. Since expP I Bp ( 0, 1f / .J Kmax) is a local diffeomorphism,
there is for each p and q sufficiently close to p and q respectively a unique
vector V E Tp•Mn close to V E TpMn such that expp (V) = q. (Here we
have identified TpMn ~!Rn~ Tp•Mn.) The path {3* : [O, 1] ----+Mn defined
by {3* ( s) ~ expp* ( s V*) is the unique geodesic near (3 joining p* and q*. DWe now anticipate Definitions B.66 and B.67 (found in Appendix B): ifk E N, a proper geodesic k-gon is a collection
r = bi : [O, Ci] ----+Mn : i = 1, ... , k}
of unit-speed geodesic paths between k pairwise-distinct vertices Pi E Mn
such that Pi = /i (0) = /i- 1 (t'i-1) for each i, where all indices are interpreted
modulo k. We say r is a nondegenerate proper geodesic k-gon if
Lp; (-"ti-1, "ti) # 0 for each i = 1, ... , k ; if k = 1, we interpret this to mean
L (r) > 0. A (nondegenerate) geodesic k-gon is a (nondegenerate)
proper geodesic j-gon for some j = 1, ... , k.
LEMMA 5.67. Let (Mn, g) be a closed Riemannian manifold with sec-
tional curvatures bounded above by Kmax > 0. Let (3 be a nondegenerate
geodesic 2-gon between p # q E Mn of edge lengths L (f3i) < 1f / .J Kmax for
i = 1, 2. Then there is a smooth nontrivial geodesic loop I on Mn with