1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

7. THE £-FUNCTION ON EINSTEIN SOLUTIONS AND RICCI SOLITONS 339

Given a path 'Y (r), its £-length is

£, (1') = [ v'i' (Rb (7) '7) +I ~;[(,}7

= [ v'i' ( R (\OT b (7)) '0) + 1(\0,), ~;1:1.}7

Let {3 (r) ~<(Jr("! (r)). Note that since g(r) = <p;go, geometrically, a point

(x, r) is the same as the point (VJr (x), 0). That is, ("! (r), r) is the same as

({3 (r), 0). We have

·. d{3 d'Y a<pr

{3 (^7 ) =;= dr = ('Pr)* dr + OT ({3 (^7 )) '

so that

Hence

L'. ("!) = fo'f VT ( R ({3 (r), 0) + l;j (r) + (grad 90 fo) ({3 (r))1;(

0

J dr

= 1

7

VT ( R ({3 (r), 0) + l(grad 90 fo) ({3 (r))l~(o)) dr

+!

7

VT (l;j (r)l

2

Jo g(O) +^2 I~ \ (r), (grad^90 fo) (!3 (r))) g(O) ) dr.

Now from (1.34) we have

(7.110) R ({3 ( r), 0) + I (grad 90 fo) ({3 ( r)) l~(o) = C

independent of {3 ( r) , and

~ (r), (grad 90 fo) ({3 (r))) g(O) = d~ Uo ({3 (r))) ·

Hence, letting (}' = 2y17 and a ( (}') ~ {3 ( (}'^2 / 4) and integrating by parts, we

have an alternate formula for the £-length on a steady gradient Ricci soliton.

LEMMA 7. 71. On a steady gradient Ricci soliton Re (go) + 'V'V Jo = 0,

we have

£,( 'Y) = ~67^312 + 2y'j' fo (a ( 2v'T)) + 1'./¥ (I~: (a {

0

) - /o (a (a))) da,

where C is given in (7.110), a((J') = <fJu2; 4 ('Y (~

2
)) and <(Jr is defined by
(7.109).
Later we shall see that the shrinking case gives us a more explicit for-
mula.
Now we compute the geodesic equation by taking a variation Y =<Sa of
a which vanishes at the endpoints. We have