1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. THE £-LENGTH AND THE £-DISTANCE 289

by reversing time. In particular, if w < +oo, let r ~ w-t, so that (N, h (r))

is a solution to the backward Ricci fl.ow on the time interval (0,w - a).^4

2.1. Space-time motivation for the £-length. We begin by moti-

vating the definition of the £-length for the Ricci fl.ow as a renormalization of

the length with respect to Perelman's potentially infinite Riemannian metric

on space-time. Given NE N, define a metric on N ~ Nn x sN x (0, T) by

(7.11) h ~ hijdxidxj + rha(:JdyadyfJ + (~ + R) dr^2 ,

where hafJ is the metric on SN of constant sectional curvature 1/ (2N) and

R denotes the scalar curvature of the evolving metric h on N. Here we have

used the convention that {xi} ~=l will denote coordinates on the N factor,

{ya}~=l coordinates on the sN factor, and x^0 ~ r. Latin indices i,j, k, ...

will be on N, Greek indices a, f3, 'Y, ... will be on SN, and 0 represents the
(minus) time component. Choosing N large enough so that f-r + R > 0
implies that the metric h is Riemannian, i.e., positive-definite. In local
coordinates,

(7.12)

(7.13)

(7.14)

(7.15)

hij = hij,

ha(:J = rha(:J,

• N
  hoo = 27 +R,

hiO = hia = hao = 0.

Let i'(s) ~ (x(s), y(s), r(s)) be a shortest geodesic, with respect to the
metric h, between points p ~ (xo, yo, 0) and q ~ (x1, y1, rq) EN. Since
the fibers SN pinch to a point as r ---+ 0, it is clear that the geodesic ')'(s)
is orthogonal to the fibers SN. (To see this directly, take a sequence of
geodesics from Pk ~ (xo, y1, 1/k) to q and pass to the limit as k ---+ oo.)

Therefore it suffices to consider the manifold .fl~ N x (0, T) endowed with

the Riemannian metric:

(7.16) h ~ hijdxidxj + ( ~ + R) dr^2.

(This metric is dual to the metric considered in [100].) For convenience,

denote x(s) ~ "((s).

Now we use s = r as the parameter of the curve. Let ~ ( r) ~ ~~ ( r).

The length of a path 1 ( r) ~ ( "!( r), r) , with respect to the metric h, is given

by the following:

(^4) we shall consider the case where a = -oo (in which case we define w - a ~ +oo).

On the other hand, if w = +oo and a= -oo, we may simply take T = -t. However, for

the backward Ricci fl.ow we are not as interested in the case where w = +oo and a> -oo.