1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. THE £-LENGTH AND THE £-DISTANCE 293

2.3. The £-distance function. Just as for the usual length functional

(perhaps it is better to compare with the energy functional), one gives the

following definition.

DEFINITION 7.9 (£-distance). Let (Nn, h (T)), TE (A, Sl), be a solution

to the backward Ricci fl.ow. Fix a basepoint p E N. For any x E N and

T > 0, define the £-distance by

L (x, T) ~ Lfp,O) (x, T) ~ i~f £ ('/'),

where the infimum is taken over all C^1 -paths 1 : [O, T] --+ N joining p to x

(the graph 1 joins (p, 0) to (x, T)). We call an £-length minimizing path a

minimal £-geodesic. We also define

(7.22) L -(x, T) =;=. L(p,O) -h (x, T) =;=. 2v:TL (x, T).

Note that the £-distance defined above may be negative. To help the

reader have a feeling for the £-distance function, we present some exercises.

EXERCISE 7 .10 (Scaling properties of £ and L). Let (Nn, h ( T)) be a

solution to the backward Ricci fl.ow, 1 : [Ti, T2] --+Na C^1 -path, and c > 0

a constant. Show that for the solution h (f) ~ ch (c-^1 f) and the path
:Y: [cT 1 , cT 2 ]--+ N defined by :Y (f) ~ 1 (c-^1 f), we have

£1i (:Y) = vc.Ch (1).

Consequently,

L~,O) (q, f) = vcLfp,O) (q, c-^1 f).

EXERCISE 7.11 (£ and L on Riemannian products). Suppose that we

are given a Riemannian product solution (N{!'^1 x N:f'^2 , hi ( T) + h2 ( T)) to the

backward Ricci fl.ow and a C^1 -path 1 = (a, {3) : [Ti, T2] --+Ni x N2. Show

that

Hence

L(;~~^2 ,o) (q1, q2, T) = L(;1,0) (q1, T) + L(;2,0) (q2, T).

It is useful to keep in mind Euclidean space as a basic example; more

generally we have

EXERCISE 7 .12 (£-distance for Ricci fiat solutions). Let (Nn, h ( T) = ho)

be a static Ricci fl.at manifold and let p EN be the basepoint. Show that

given any q EN and f > 0, the £-length of a C^1 -path 1: [O, f] --+ N from

p to q is

£ (1) = 12V'f I~; (cr2/4)12 dcr,

which is the same as (7.2). Hence a minimal £-geodesic 1 is of the form

(7.23) 1 (T) = {3 (2v:T),