1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

102 2. KAHLER-RICCI FLOW

P(r) on en{O} for those whose behavior at the boundary izi = 0 implies
that

l_: the metric is completed by adding a smooth point at r = -oo;

2_: the metric is completed by adding an orbifold point at r = -oo;

3_: the metric is completed by adding a JP'n-l at r = -oo; or

4_: the metric is complete as r--+ -oo;

and whose behavior at the boundary lzl = oo implies that

1+: the metric is completed by adding a smooth point at r = +oo;

2+: the metric is completed by adding an orbifold point at r = +oo;

3+: the metric is completed by adding a JP'n-l at r = +oo; or

4+: the metric is complete as r--+ +oo.

Of course, not all combinations of these alternatives are globally com-

patible. For example, it is easy to see that the growth condition I.Pr > 0

prohibits completing the metric by adding a JP'n-l at izl = 0 and a smooth

point at izl = oo. Nonetheless, this has been a productive line of research.

In the remainder of this section, we will survey some of its results.

It is possible to add a smooth point at izJ = 0 and to construct a unique

steady Kahler-Ricci soliton on en that is complete as Jzl --+ oo. In complex

dimension n = 1, this is just the cigar soliton discovered by Hamilton and

discussed in Chapter 2 of Volume One; the examples in higher dimensions

are due to Cao [47]. These solitons have the following asymptotic behavior:

in the sphere s^2 n-l at metric distance r » 0 from izl = 0, the Hopf fibers

U(l) · z have diameter 0(1), while the JP'n-l direction has diameter 0( ylr).

Accordingly, one calls this cigar-paraboloid behavior.

It is also possible to add a smooth point at izJ = 0 and to construct ex-

panding Kahler-Ricci solitons on en that are complete as lzl --+ oo. There

is in fact a 1-parameter family (en,ge)e>o of such examples in each dimen-
sion, due to Cao [48]. (It is a heuristic principle that expanding solitons are
easier to find than their shrinking cousins. For these examples, satisfying
the boundary condition at izl = 0 reduces the parameter space by one di-
mension, but completion as I z I --+ oo comes for free.) Each soliton (en, ge)
is asymptotic as lzl--+ oo to the Kahler cone (en{O},§e), where the metric
.Ye is induced by the Kahler potential P(r) = e^6 r ;e.
For each k = 2, 3, ... , the authors of [142] add an orbifold point at
izl = 0 and a JP'n-l at izl = oo to construct a unique shrinking Kahler-Ricci
soliton on an orbifold, which is called !:Ff:. The compact orbifold (J'f: may be
regarded as JP'n /Zk branched over the origin and the JP'n-l at infinity. The

orbifold singularity at the origin is modeled on en /Zk·

For each dimension n 2: 2 and k = 1, ... , n-1, the authors of [142] add

a k-twisted JP'n-l at izl = 0 and construct a unique shrinking Kahler-Ricci

soliton metric that is complete as izl --+ oo. The resulting soliton has the

topology of the complex line bundle e '----* L:!:_k -+> JP'n-l characterized by

(c1, [L::]) = -k, where c1 is the first Chern class of the bundle and L: ~ JP'^1
is a positively-oriented generator of H2(lP'n-1; Z). (For example, the total