1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. THE SECOND VARIATION FORMULA FOR LAND THE HESSIAN OF L 317

we have

1

7

2y'T IVxY +Re (Y)l^2 dr

= 1

7

2y'T [vxY +Re (Y) - 2 ~Y[

2

dr + ~ IYl

2

1:,

where we integrated by parts. Note that the assumption of the lemma

implies that Y (a;) is smooth in O" and lim 7 -+0 )T IY (r)l^2 = 0. The lemma

now follows from (7.64). D

We now consider a special case of this formula. As above, let "! : [O, f] ~

M be an £-geodesic. Fix a vector Y 7 E T"f(i')M and define a vector field

Y ( r) along "( by solving the following ODE along -ry:

(7.66)

1

VxY = -Rc(Y) +

27

Y, r E [O,f],

Y (f) = Y7.

Note that any vector field along "( can be considered as a variation vec-

to~ field. In particular, we may extend Y ( r) to Y ( r, s) for some smooth

variation of ry.

REMARK 7.38. Equation (7.66) is equivalent to

which essentially says )TY is parallel with respect to the space-time con-

nection. Note that if "is : [O, f] ~ M is a 1-parameter family of paths such

that X = g 7 "(s and Y = gs "(s, then [X, Y] = 0 and (7.66) may be rewritten
as

( \7 X +Re-

2

~g) (Y) = 0,

which is reminiscent of the gradient shrinker equation.

From (7.66) we compute

(7.67) !!:._ dr IYl·^2 = dd r [g (Y, Y)] =^2 (\7 x Y, Y) +^2 Re (Y, Y) = r !. IYl^2.

Solving this ODE, we have

(7.68) IY (r)I^2 =^7 ~ IY (f)I^2 ·

7

Thus Y (0) = 0. Hence by (7.65) we have the following.