1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

8. .C-JAOOBI FIELDS AND THE £-EXPONENTIAL MAP 345

On the other hand, taking I to be a minimal £-geodesic with I (f) = q,^13

we have

f (I (f) 'f) = f, (I (f) 'f) -

1

~ r VT l'Y (T) - \i'f (I (T), T)l~(r) dT

2yT lo

::; £(1(f) ,f).

Since I (f) = q is arbitrary, we conclude that f = f, on N x (0, oo) for f

defined by (7.118).

EXERCISE 7.81. Let (Nn, h (t)), -oo < t < 1, be a shrinking gradient

Ricci soliton in canonical form and consider (N, h ( T)) , T E ( O, oo) , where

h (T) ~ h (1-T). Show that for any p, q E N and f > 0, if we take any
sequence Ti ---+ 0 and minimal £-geodesics Ii : h, f] ---+ N with Ii (Ti) = p
and Ii (f) = q, then a subsequence /i converges to a minimal £-geodesic
/ : (0, f] ---+ N with / (f) = q. In particular, independent of the choice of
p EN, for any path (3: (0, f]---+ N with/ (f) = q, we have

£ ((3) 2: £ (I).

8. £-Jacobi fields and the £-exponential map

Continuing our mimicry of Riemannian comparison geometry, in this
section we derive the £-Jacobi equation, which is a linear second-order ODE
along an £-geodesic, and we derive an estimate for the norm of an £-Jacobi
field. We also discuss the £-exponential map, its Jacobian, called the £-
Jacobian, and briefly mention the £-index lemma. These results will be of
crucial importance to our discussion of the reduced volume and its applica-
tions in the next chapter.
· Throughout this section (Mn, g ( T)) , T E [O, T] , will denote a com-
plete solution to the backward Ricci flow satisfying the curvature bound
max {IRml, IRcl}::; Co < oo on M x [O, T], and p EM is a basepoint.
Before discussing Ricci flow, we first recall some basic Riemannian ge-
ometry that is relevant to the material in this section. A good reference for
comparison Riemannian geometry, besides Cheeger and Ebin [72], is Mil-
nor's book [265]; in our setting, the Riemannian path energy is analogous to

the £-length. Let I : [a, b] ---+ (kt, g) be a unit speed geodesic, let ,-Y ~ ¥a,

and let ds denote the arc length element. The index form is defined by

(7.119) I(V,W)~ 1b((V'-yV,V'-yW)-(R(V,,-Y),-Y,W))ds,

where V and W are vector fields along I perpendicular to ,-Y and vanishing
at the endpoints. By the second variation of arc length formula, under these
assumptions on V and W, we have

oi,wL (1) =I (V, W)'

(^1) 3see the exercise below.