100 2. KAHLER-RICCI FLOW
and
32
Ra73 = - f)zaf)zf3 log <let g,
exactly as in (2.6). If Q denotes the soliton potential function with gradient
vector field X, then equation (2.96) reduces to
a2
azaf)zf3 (log det g - Q - AP) = 0.Because we are interested in shrinking solitons, we may assume that A< 0.
Then (modifying the Kahler potential P by an element in the kernel of V''\7
if necessary) we may assume that Q = log det g - AP. This will give us a
soliton provided X = grad Q is holomorphic, that is, provided that
o-- ~xf3 a-za -- ~ f)zO!. ( 9 f31~Q) {)z'Y.
Substituting Q = log <let g - AP, we obtain a single fourth-order equation
for the scalar function P, namely
(2.97) a~a [gf31 f)~'Y (log det g - AP)] = 0.
To proceed, we adopt the Ansatz that the potential Pis invariant under
the natural U(n) action in the sense that it is a function of r =log 2::::~= 1 lzal^2
alone. In this case, setting cp =Pr, we have
(2.98) 9a{3 = e-r cp8a(3 + e-^2 r ('Pr - cp )za z!3,
so our P will be a Kahler potential if and only if cp and 'Pr are everywhere
positive. Now (following [47] and [142]) we can write (2.97) as the fourth-
order ODE
( 2.99 ) Prrrr - 2--p P:jrr + nPrrr - ( n - l ) p:r p 2 + /\ '(P rrr p r - Prr 2) = 0.
rr r
We shall see that only two of the four arbitrary constants in its solution are
geometrically significant.
To simplify (2.99), notice that xa = ga73 8 ~(3 Q = ~;,. za will be holomor-
phic if and only if Qr= μPrr for someμ ER Since g will be Kahler-Einstein
if X = 0, we may assume that μ !-0. Substituting Q = log det g - AP, one
then obtains(log 'Pr )r + (n - l)(log cp )r - μcpr - Acp - n = 0,
which is a second-order equation for cp, hence a third-order equation for P.
(This integration can also be accomplished by standard ODE techniques.)Because 'Pr > 0 everywhere, one may regard r as a function of cp and hence
may write 'Pr F(cp). One finds (remarkably) that F satisfies a linear
equation
n-l
F' + (---μ)F-(n + Acp) = 0,
cp