- EXISTENCE OF AN ASYMPTOTIC SHRINKER 107
5.1. Two estimates for the reduced distance function.
In this subsection we discuss two estimates for the reduced distance
function which we shall use in the proof of the existence of an asymptotic
shrinker.
5.1.1. A review of the reduced distance.
Let (Mn, g (T)), T E [O, T), where 0 < T _:::; oo, be a complete solution
to the backward Ricci fl.ow on a connected manifold. The genesis of Perel-
man's £-geometry is the notion of the £-length of a piecewise C^1 -path
"(:[Ti, T2] -t M, where [Ti, T2] C [O, T), which is defined by
(19.43) .c b) ~ 1
72VT (R b ( T) 'T) + I ~"( ( T) i 2 ) dT.
Tl T g(T)
Fixing a point p EM, we define the L-distance from a point (x, T) E
M x (0, T) to the 'base point' (p, 0) by
(19.44) L (x, T) ~ L(p,O) (x, T) ~ i~f .C ("!),where the infimum on the RHS is taken over all piecewise C^1 -paths 'Y
[O, T] -t M joining p to x.^26 This space-time distance-type function is
fundamental to the study of the Ricci flow.
The reduced distance from (x, T) to (p, 0) is defined by
1
(19.45) £ (x, T) ~ 2./TL (x, T).
5.1.2. A lower estimate for the sum of£ at two different points at the
same time.
We have the following estimate of Perelman relating the reduced distance
at two different points at the same time (see Lemma 2.2 in Ye [191], (39.6)
in Kleiner and Lott [110], or Lemma 9.25 on p. 192 ff. in Morgan and Tian
[133]).
LEMMA 19.46. For any rt,-solution (Mn, g (t)), t _:::; O, and basepoint
(p, 0), we have that the reduced distance of g (-T) satisfies
c (n) d;(r) (q1, q2)(19.46) -£ (q1, f) _:::; £ (q2, f) + 1 - -
Tfor all f E (0, oo) and q1, q2 EM, where c (n) = 216 (~+ 2 ) 2 • In other words,
c (n) d;(r) (q1, q2)
£ ( q1' f) + £ ( q2' f) ~ T - - 1.(^26) More generally, for points p, q E M and times 0 :::; 71 < T2 < T, we define the
£-distance by
L(p,r 1 ) (q, 72) = i~f £ ('Y),
where 'Y: [71, 72] --+ M joins p to q. If 71 ~ T2, then we define L(p,ri) (q, 72) = oo.