1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

We compute

b.gf(x)

n-l

1. ESTIMATES FOR CHANGING DISTANCES

= L V'V' f (Ei, Ei) + V'V' f (1^1 (so),1' (so))

i=l

= n-l tt 8r2 EJ2 I r=O f (expx (rEi)) + 8r2 [J2 ' r=O f (expx (r1' (so)))

n-l 321 321

= tt 8r2 r=O Lg ( lTK) + 8r2 r=O (so + r)

~ f ( (n - 1) {(' (s))

2

- (^2 ~ (R ('y'(s), E,(s)) E,(s), J(s))) ds,

45

where we used (18.7) to obtain the last equality. The lemma now follows

from (18.6) and

n-l

L ( R ( 1' ( s), Ei ( s)) Ei ( s), 1' ( s)) = Re ( 1' ( s), 1' ( s)).

i=l

0

1.3. Lower bounds for the time derivative of distance.

We now prove the main result of this section, which is Lemma 8.3 in
[152]. In what follows, gt may denote the lim inf of either forward or
backward difference quotients.

THEOREM 18.7 (Estimate for changing distances). Let (Mn,g(t)), t E

[O, T), be a complete solution to the Ricci fiow,

(1) (Heat-type inequality for distance function) Let (xo, to) E M x

[O,T). If

(18.8)

Rc(y, to) ::; (n - l)K for ally E Bg(to) (xo, ro),

where K 2:: 0 and r o > 0, then for all x E M - Bg(to) (xo, ro) the

distance function is a supersolution to the heat-type equation

( 8

8

• b.g(t)) dg(t)(x, xo)I 2:: -(n - 1) (~Kro + ]:_).
  t t=to 3 ro
  This inequality is understood in the barrier sense.^2
  (2) (Changing distances under Ricci flow) Let to E [O, T) and let xo, x1
  E M be two points such that

Rc(x, to) ::; (n - l)K for all x E Bg(to) (xo, ro) U Bg(to) (x1, ro)

for some K 2:: 0 and ro > 0.

(^2) For differential inequalities in the barrier sense for the reduced distance function,
see Chapter 7 in Part I. '