- ESTIMATES FOR CHANGING DISTANCES 41
denote the lim inf of forward difference quotients. We have a similar defini-
tion for^8 a/ (x, t), the lim inf of backward difference quotients, with t + h
replaced by t - h.
Let (Mn, g (t)), t E [O, T), be a smooth 1-parameter family of complete
metrics. Given to E [O, T), let Z (to) denote the set of all unit speed minimal
geodesics joining xo to x, with respect to g (to). Note that Z (to), with the
subspace topology induced by the compact-open topology on the space of
continuous paths, is compact and nonempty.
The evolution of the distance function is given by the following.LEMMA 18.1 (Time derivative ~~ of the distance function). For to E
[O, T), in the sense of the lim inf of forward difference quotients~-1 d 9 (t)(x,xo) = min (- 2
1
1 ~g (to) (r/ (s) ,r/ (s)) ds).
ut t=to 'T/EZ(to) 'f/ utPROOF. For to E [O, T) and each 'T/ E Z (to) (see Lemma 3.11 in Volume
One)(18.1) !I Lg(t) (TJ) = ~ 1 ~~(to) ('TJ' (s) ,'T/^1 (s)) ds,
t=to 'f/where Lg(t) (TJ) is the length of the path 'T/ with respect tog (t) and where s
denotes arc length with respect to g (to). For each 'TJ E Z (to)
8t f)-1 t=to dg(t)(x,xo)::::; dt di t=toLg(t) (TJ),
where we have used dg(t)(x, xo) ::::; Lg(t) (TJ) fort~ to and we have also used
dg(to)(x, xo) = Lg(to) (rJ). Therefore
~-1 dg(t) (x, xo) ::::; min (- 2
1
1 ~g (to) ('TJ' (s), rJ
1
(s)) ds).
ut t=to 'T/EZ(to) 'f/ utNow we prove the reverse inequality. Suppose for i E N that hi ---+ o+
and 'T/i is a unit speed minimal geodesic joining xo to x, with respect to
g (to+ hi)· Passing to a subsequence, we have 'T/i---+ 'TJ 00 , where 'T/oo is a unit
speed minimal geodesic joining xo to x, with respect tog (to). Then
dg(to+hi) ( x, xo) - dg(to) ( x, xo) > L 9 (to+hi) ( 'T/i) - Lg( to) ( 'T/i)
hi hi= :t I Lg(t) (rJi)'
t=to+kiwhere 0 < ki <hi, by the mean value theorem. Taking the limit as i---+ oo
and since hi ---+ o+ is a subsequence of an arbitrary positive sequence tending