1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. FIRST VARIATION OF £-LENGTH AND EXISTENCE OF £-GEODESICS 305

curvature. Given a space-time point (p, T1) E M x [O, T) and a tangent

vector VE TpM, there exists a unique £-geodesic I : [T1, T) -+ M with

lim vfrdd"f ( T) = V

T---+T1 T

NOTATION 7.26. We shall usually denote I as IV·

PROOF. The main idea of the proof is to change variables from T to

O" and to apply standard ODE theory. In local coordinates, the £-geodesic

equation (7 .35) for (3 ( O") = I ( 0"^2 / 4) is

(7.49)

The above system of ODE is of the form

d2(3 (d(3 )

d0"2 = F dO"' (3, (} '

where F is a smooth function. From elementary ODE theory, given any point
p EM and tangent vector VE TpM, there exists a path
(3 : [2JT1, 2JT1 + c] -+ M
solving (7.49) with ~~ (2JT1) = V for some E > 0.
We prove by contradiction that (3 can be extended to an £-geodesic
on [2y'Tl, 2VT). Suppose the maximal time interval of existence of (3 is
[2y'Tl, 2y'T2) for some T2 < T. By Lemma 7.24, we have I~~ I ::; C. This
g(a^2 /4)
implies (3 ( O") converges as O" -+ 2y'T2 to some q E M using the assumption
that the solution g ( T) is complete. Hence we can extend (3 beyond 2.JT2.
This is a contradiction and the lemma is proved. D

We end this section by proving a lemma about the existence of £-

geodesics between any two space-time points (this may also be proved with-

out using the calculus of variations).

LEMMA 7.27 (Existence of minimal £-geodesics). Let (Mn,g (T)), TE

[O, T], be a complete solution to the backward Ricci fiow with bounded sec-

tional curvature. Given p, q E M and 0 ::; T1 < T2 < T, there exists a

smooth path I ( T) : h, T2] -+ M from p to q such that I has the minimal

£-length among all such paths. Furthermore, all £-length minimizing paths

are smooth £-geodesics.

PROOF. Here we sketch a proof using the direct method in the calculus

of variations. Let Ii : [Ti, T2] -+ M, i E N, be a minimizing sequence for

the £-length functional. That is, £(1i)-+ inf.yC(i') as i-+ oo, where the
infimum is taken over all C^1 -paths i': [Ti, T2] -+ M from p to q. We have

1

72

vrldli (7)1

2

dT::; £(1i)-~3 (T~/

2

-Tt^12 ) inf R(x,T)::; c.

71. dT g(T) Mx[T1,T2]