1547671870-The_Ricci_Flow__Chow

308 B. SOME RESULTS IN COMPARISON GEOMETRY

In fact, Ts E (0, £), because a (0), a(£) E JH['Y. Thus we can apply the first

variation formula to conclude that a (Ts) l_ /3s (0) at a (Ts) = f3s (0). Now

parallel translate the unit vector a (Ts) to get a unit vector field U along f3s,

and define the variation vector field

0 :S a :S L (f3s).

Note that V is the Jacobi field of a family of geodesics joining a to / (s);

in particular, v (f3s (0)) = a (Ts) and v (f3s (£s)) = o. Because f3s is minimal

among all geodesics joining a to r ( s), applying the second variation formula

with S ~ (f3s)* (d/da) yields

f L(f3s) ( 2 )

0 :S lo IV sVI - (R (S, V) V, S) da.

Since \7 sV = -L(1s) U, we have

1

L(f3s) 2 1 1L(f3s) 1

IVsVI da= 2 da= L(f3).

o L(f3s) o s

But since sect (g) > 0, there is E > 0 depending only on a c Mn such that

f L~) fl

lo (R (S, V) V, S) da ?_lo (R (S, V) V, S) da

fl (L (f3s) - a)2 d

• E lo L (f3s) a
  L (f3s) - 1
  ?_E L(f3s).

Hence JH['Y can fail to be totally convex only if

1 L (f3s) - 1

O :S L (f3s) - E L (f3s) '

which is equivalent to the condition

E(L (f3s) - 1) :S 1

holding for all s ?_ so > 0. Since L (f3s) ---+ oo as s ---+ oo, this is impossible.

D

PROOF OF PROPOSITION B.54 IN THE CASE sect (g) ?_ o. If JH['Y is not

totally convex, there are x, y E JH['Y and a unit-speed geodesic a : [O, £] ---+Mn

such that a (0) = x and a(£) = y, but a</:. JH['Y. As in the proof above, there

is so> 0 such that anB(r(s),s) I- 0 for alls?_ so. Since a([O,£]) is

compact, there are for any s ?_ so some Ts E [O, £] and a minimal geodesic

f3s,Ts : [O, £s,Tsl ---+ Mn such that f3s,Ts (0) = a (Ts) and f3s,Ts (£s,T.) = I (s ),

where £s,Ts ~ L (f3s, 7 .) = d (a,/ (s)) < s. Define

E ~ so - d (a ( Ts 0 ) , / (so)) = so - d (a, / (so)) > 0