1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

42 18. GEOMETRIC TOOLS AND POINT PICKING METHODS

to 0, we have

a-1 at t=to dg(t)(x,xo) 2: dt di t=to Lg(t) (rJoo)

2: min ( ~ 1 ~g (to) ( rJ^1 ( s) , rJ^1 ( s)) ds).

'f/EZ(to) 2 'f/ ut

This completes the proof of the lemma. D

In the special case of the Ricci fl.ow we immediately obtain the following.

COROLLARY 18.2 (Time derivative of distance under Ricci fl.ow). Let

(Mn, g (t)), t E [O, T), be a complete solution to the Ricci flow. For to E

[O, T), in the sense of the liminf of forward difference quotients,

(18.2) ~-, dg(t)(x,xo) = min (-1Rc 9 (to) (rJ' (s) ,rJ^1 (s)) ds).

ut t=to 'f}EZ(to) 'f/

MINI-PROBLEM 18.3. Is it true that under the Ricci flow,

~-1 d 9 (t) (x, xo) = -1 Rc 9 (to) ( 1' (s), 1' (s)) ds

ut t=to 'Y

for any 1 E Z (to)?

Although the above discussion is sufficient for our needs, we recall the
lower bound for the time derivative of the distance function using the lim inf
of backward difference quotients, in view of the simplicity of its derivation.

LEMMA 18.4 (Time derivative ~t ofthedistancefunction). If(Mn,g(t)),

t E [O, T), is a smooth 1-parameter family of complete metrics, then for
to E [O, T) we have

(18.3) c:;-1 dg(t)(x,xo) 2: -

2

1

max (1 ~g (to) (rJ'(s),rJ'(s))ds)

ut t=to 'f/EZ(to) 'f/ ut

in the sense of the liminf of backward difference quotients.

PROOF. Let rJ: [O, so] ---+ M be any unit speed minimal geodesic joining
xo to x, with respect to g (to). We have

L 9 (t) (rJ) 2: d 9 (t)(x,xo),

L 9 (to) (rJ) = d 9 (to)(x, xo)

fort E [O, T). Therefore we have

~~I d 9 (t)(x, xo) 2: : I L 9 (t) ('r/).

t=to t t=to

Taking the maximum over minimal rJ, we have

~-1 d 9 (t)(x,xo)2: max ddl Lg(t)(rJ),

ut t=to 'f}EZ(to) t t=to