- EXAMPLES OF KAHLER-RICCI SOLITONS 101
whose solution iscpr = cp^1 -neμ,r.p(v +>..In+ nin-1),
where v is another arbitrary constant and In = J cpne-μc.p dcp. This leads to
n-1
111 = v111^1 -neμc.p - ~111 - >.. + μ ~ n! μj111j+l-n
rr r r l+n L....J ·1 r '
μ μ j=O J·which is a separable first-order equation for cp. The third and fourth ar-
bitrary constants arise when one finds the implicit solution r( cp) and then
integrates cp to obtain P. These are geometrically insignificant, because they
disappear in (2.97); the two geometric degrees of freedom are the parameters
μ =J 0 and v.
In summary, what we have thus far accomplished is to construct a po-
tential function for a U(n)-invariant Kahler-Ricci soliton metric (2.98) on
en\ { 0}' hence a possibly incomplete Kahler-Ricci soliton metric on F'f:.
What remains is to choose μ and v in order to get a complete metric on :F'f:.
Our choices of the two parameters will be determined by the two boundary
conditions as r----+ ±oo.
Since cpr > 0, we may define a < b E [O, oo] by a = limr->-oo cp(r)
and b = limr->oo cp(r). If a > 0, one may write P(r) = ar + p(ear) in
a neighborhood of [z[ = 0, with p smooth at zero, p(O) = 0, p'(O) > 0.
Similarly, if b < oo, one may write P(r) = br + q(e-f3r) in a neighborhood
of [z[ = oo, with q smooth at zero, q(O) = 0, q'(O) > 0. One then takes
advantage of the following observation.LEMMA 2. 77 ( Calabi). Assume that k > 0 and a, b E (0, oo).
(1) When a = k, the potential P(r) = ar + p(ear) induces a smooth
Kahler metric on a neighborhood of So in :F'f:. Any IP'^1 in So has
area a7r.
(2) When f3 = k, the potential P(r) = br + q(e-f3r) induces a smooth
Kahler metric on a neighborhood of 800 in :F'f:. Any lP'^1 in 800 has
area b7r.For the proof, see [44] or [142, Lemma 4.2].
With more work, one finds that it is possible to satisfy both boundaryconditions by appropriate choices ofμ and v. (See [47] or adapt the argu-
ments in [142, §4.1].) Normalizing by fixing >.. = -1, these choices yield
a = n - k and b = n + k. Since one needs a > O, one obtains a unique
gradient shrinking Kahler-Ricci soliton on :F'f: for each k = 1, ... , n - 1.
These are the Koiso solitons.7.3. Other U(n)-invariant solitons. The construction we have de-
scribed in Section 7.2 above has natural generalizations allowing the discov-
ery of other explicit Kahler-Ricci soliton examples. Namely, one searches
the 2-dimensional (μ, v) parameter space of U(n)-invariant Kahler potentials