400 8. APPLICATIONS OF THE REDUCED DISTANCE
we have l·l~(T) s eCoT l·l~(To) = e^00 T l·l~(o) acting on vector fields for T E
[To - T /2, To]. We can estimate [, (f3) as follows:
[, (f3)S {To VT (Rg( 7 ) (f3(T))+Id1
2
) dT
~-T~ TgWs -2 ( T3/2 3/2) GT 1To. ;::; I d/312
0 - (To - T /2) sup Rg(t) + e(^0) v T -d dT
3 Mx[O,T/2] To-T/2 T g(O)
= ~ (T~/2 - (To -T/2)3/2) ( sup R9(t) + 4eCoT (d9(0) (q, qo))2)
3 Mx[O,T/2] T^2~ T3/2 ( G 4eCoT r5)
s3 no+ T2.Let a : [O, To - T /2] --+ M be a minimal £-geodesic with a (0) =Po and
a (To -T/2) = qo. Then
£(a)= 2VTo -T/2 · £ (qo, To -T/2) s nylT/2.
Consider the concatenated path:'Y (T) =(a'-" f3) (T) = { a (T) if t E [O, To -T/2],
. f3(T) iftE[To-T/2,To].
This path is well defined and piecewise smooth. We have
1 1
£ (q, To) s f1f!"£ (!') = fFT1""" [£(a)+£ (f3)]
2vTo 2vTos ~ [nVT/2 + ~T
3
12
( nCo +4e~~r5) J
~ C1 (ro, n, T, sup. Re g(t)).
Mx[O,T/2](ii) Choosing ro = ri in (i), we have
V (To)= JM (47rTo)-nl^2 e-£(q,To)dμg(O) (q)
2: { (47rTo)-n/2 e-£(q,To)dμ9(0) (q)
j B9(0) (qo,ri)2: { (47rTo)-n/2 e-C1dμ9(0) (q)
} B9(0) (qo,r1)2: v1 ( 47rT)-nl^2 e-Ci ~ C2 (v1, ri, n, T, sup Re g(t)).
Mx[O,T/2]
D