1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. FIRST VARIATION OF £-LENGTH AND EXISTENCE OF £-GEODESICS 301

EXERCISE 7.22. Estimate the speed of an £-geodesic for a solution

(Mn, g ( T)) , T E [O, T), to the backward Ricci flow with

c

IRm (x, T)I::::; T- 7 on M x [O, T).

Hint: What does the Bernstein-Banda-Shi estimate say about IV RI?

3.3. Space-time approach to the £-geodesic equation. We now

compare the £-geodesic equation for 'Y with the geodesic equation for the

graph i (T) = ("! (T); T) with respect to the following space-time connection

(see also Lemma 4.3 in [100]):

(7.39)

(7.40)

(7.41)

(7.42)

where i,j, k 2".. 1 (above and below), and the rest of the components are

zero. It i~ instructive to compare the Christoffel symbols f' above with the
symbols N f' of the Levi-Ci vita connection N fJ for the metric h introduced in
Exercise 7.4. Fork 2': 1, note that f'~b = Nf'~b is independent of N, whereas

-o fab = hmN-+oo · N-0 fab for all a,b 2': 0.

Let T = T(o-) ~ o-^2 /4, i.e., o-~ 2y!T. We look for a geodesic, with respect
to the space-time connection defined above, of the form

P(o-) ~ ('Y(T(o-)), o-^2 /4),

where 'Y : [Ti, T 2 ] ----+ M is a path. For convenience, let f3(o-) ~ "f(T(o-)),
pi~ xi o f3 ~ f3i for i = 1, ... ,n, and p^0 ·~ x^0 op (so that p^0 (a-)= o-^2 /4).
By direct computation, we have

df3k O" d"(k

do- 2 dT '

dpo O"

do- 2'

and

We justify the change of variables from T to o-via the geodesic equation with
respect to f' by showing that the time component of P satisfies the geodesic