- EQUATIONS AND INEQUALITIES SATISFIED BYLAND .e 329
SOLUTION TO EXERCISE 7.55. We compute, using (7.94),-^8 · ( 8£ £ d1 )
8
(2yrf (r (T), T)) = yT 2-
8+ - + 2\7£ · -
T T T ~=VT ( R+ l~;I' -Ive-~I')
The next exercise· characterizes when an integral curve of f is an £-
geodesic.EXERCISE 7.57 (Integral curves of \7£). Show that for any solution
(Mn, g (T)), TE [O, T), to the backward Ricci flow, a smooth integral curve
r of \7 f is an £,..geodesic if and only if g 7 (\7 £) = 0 along r, where \7 f is the
gradient vector field.
SOLUTION TO EXERCISE 7.57. In view of the £-geodesic equation
(7.32), we compute
1 1
\7 \7.e \7 f - :2 \7 R + 2 Re (\7 £) + 2 T \7 f= t\7 (1\7£1
2- R + ~) + 2Rc (\7£).
Applying the identity (7.94) for £, we obtain
1 1
\7 \7.e \7 f - :2 \7 R + 2 Re (\7 £) + 2 T \7 f= -\7 ( ~:) + 2 Re (\7 f)
= -:T (\7£)'
where \7£ is the gradient off (considered as a vector field), i.e., (\7£)i =
giJ'\7jf.
We also note the following consequence of (7.91) which we shall revisit
later when considering the reduced volume monotonicity.
EXERCISE 7.58 (Pointwise monotonicity of reduced volume integrand).
Show that for a solution g 7 g = 2 Re of the backward Ricci flow,(:T +£\7.e) ((47rT)-nf2e-.edμ)
= (-!!'_ -8£ + R .:__ l\7£12 + .6.f) (47rT)-n/2 e-.edμ
2T 8T
:S 0,where £ denotes the Lie derivative. In what sense is the inequality true?
Note that f is only a Lipschitz function. See Section 9 of this chapter for
some hints.