48 18. GEOMETRIC TOOLS AND POINT PICKING METHODS(2)(b) When dg(to) (xo, x1) < 2ro, the minimal geodesic')' (s) in (18.12)
joining xo to x1 is contained in Bg(to)(xo, ro) U Bg(to)(x1, ro). By (18.12) we
have: I dg(t)(x,xo) ~ -jRcg(to) ('Y'(s),ry'(s))ds
t t=to 'Y
~ -(n - l)K L g(to) ('Y)~ -2(n - l)Kro.
D
1.4. Estimating Re along a stable geodesic.
From (18. 7) or equation (18.14) in the proof of Theorem 18. 7 just above,
we can easily derive the following consequence. Recall that a geodesic is
stable if its second variation of arc length is nonnegative.PROPOSITION 18.8 (Estimate for Re along a stable geodesic). If (Mn, g)
is a Riemannian manifold and if 'Y: [O, L] --+ M is a stable (e.g., minimal)
unit speed geodesic withRe::; (n - 1) K in B ('Y (0), r) U B ('Y (L), r),
where K > 0, r > 0, and L ~ 2r, then
j Re ( ry
1
( s) , ry' ( s)) ds ::; 2 ( n - 1) (~Kr + ~).In particular, taking r = l/VK, we have that if
Re::; (n - 1) K in B ( 'Y (0), 1/VK) u B ( 'Y (L), 1/VK),
where K > 0 and L ~ 2/VK, then
j Rc(ry'(s),ry'(s))ds::;
1
3
° (n-l)VK.
Let (Mn, g ( r)) be a solution to the backward Ricci flow. Since( ~~ dg( 7 )) ( x, y) ::; ~ Re (/3' ( s) , /3^1 ( s)) ds
for any minimal geodesics /3 joining x to y (with respect to g ( r)), we con-
clude by Proposition 18.8 that if Re::; (n - 1) Kin B (x, r) UB (y, r), where
dg(T) (x, y) ~ 2r, then(18.15) ( ~~ dg(T)) (x, y) ::; 2 (n -1) (~Kr+~).
On the other hand, under the same assumptions as above except that
dg( 7 ) (x, y) < 2r, we have