1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. THE £-LENGTH AND THE £-DISTANCE 291

In particular, Nf'~b are independent of N, whereas

hm · N-o r .. = 0

N-+oo iJ '

hm · N-o r.

N-+oo i^0 =^0 '

N-o 1

N-+oo lim I'oo = --. 2T

2.2. The £-length. A natural geometry on space-time (in the sense of

lengths, distances and geodesics) is given by the following.

DEFINITION 7.5 (£-length). Let (Nn, h (T)) 'TE (A, 0)' be a solution to

the backward Ricci fl.ow g 7 h = 2 Re, and let 'Y : [Ti, T2] -----t N be a piecewise

01 -path,^5 where [Ti, T2] c (A, n) and T1 ~ 0. The £-length of 'Y is^6

(7.17) .c ('Y) ~ Lh ('Y) ~ 1

72

VT (R ('Y (T) 'T) +I ~'Y (T) 1

2

) dT.

'Tl T h(T)

Later we shall take T1 = 0 and call T2 = f.

REMARK 7.6. Taking T1 = 0, the subsequent degeneracy introduced by

the ft factor in (7.17) reflects the infinite speed of propagation of the Ricci

fl.ow (as a nonlinear heat-type equation for metrics). We also note the formal

similarity between R+ I~ 1

2

and the quantity R+ l'V Jl^2 which we considered

for gradient Ricci solitons and which also appeared in the definitions of

energy and entropy; this seems like more than just a coincidence.

The £-length is defined only for paths defined on a subinterval of the

time interval where the solution to the backward Ricci flow exists. Note that

.C may be negative since the scalar curvature may be negative somewhere.

This is in contrast t<;> the energy defined in Section. 1 above for a static

metric. Often we shall use the following conventions:

(7.18)

We may rewrite .C as

(7.19)

1

2y0'2 (()2 I df3

1

.C ('Y) = 4R (f3 (O") '()2 /4) + d (O")^2 ) dO".

2y0'1 (} h(<J'2/4)

This is especially useful in the case T1 = 0.

Because of the I ~:;'. 1

2

term on the RHS of ( 7 .17), .C ('Y) looks more like

an energy than a length. Another way to obtain .C, which is related to the

(^5) That is, ry ( ":) is a 01 function of a.
(^6) R (ry (7), 7) is just a notation meaning Rh(r) (ry (7)), where Rh(r) is the scalar cur-
vature of (Nn, h (7)).