1547671870-The_Ricci_Flow__Chow

2. ETERNAL SOLUTIONS 25

induced by the metric. One can also see the cigar's asymptotic approach to
the cylinder in another way: the geodesics in the cigar metric are exactly
those curves "'( ( t) = ( r ( t) , e ( t)) given in polar coordinates by solutions to
the ODE system

e" +

2

e' r' = 0

r (1 + r^2 )

r" - 1: r2 [ (r')2 + (e')2] = 0.

From this, it is not hard to see that e (t) can be written in the form

e ( t) = a + bt + bit r2d~T)

where a, b E JR are arbitrary constants. In particular, when ro = r (0) is
large, a Euclidean circle of radius ro in JR^2 is close to being geodesic for a
short time.

REMARK 2.5. The cigar soliton is actually a Kahler metric

dz dz

9"'£, = 1 + lzl2

on C ~ JR^2 , and hence a Kahler-Ricci soliton. In fact, the cigar is the simplest
representative of an entire family of Kahler-Ricci solitons that exist on c^2 m
for all m EN and on certain other complex manifolds. Various other metrics
in this family were introduced in the papers [88], [23, 24], and [40]. See
also [77].

is

REMARK 2.6. The cigar metric on the topological cylinder

C = {(x,y): x E JR and y E JR/Z}

dx^2 + dy^2

1 + e-2x.

The details are left as an easy exercise for the reader.

To write the cigar metric in its natural coordinates on JR^2 , define

s ~ arcsinhr =log (r + ~).

It is easy to see that s represents the metric distance from the origin. Indeed,

since

dr = cosh s ds = J 1 + r2 ds,

formula (2.5) becomes

(2.7) g"'E, = ds^2 + tanh^2 s dB^2.

Then formula (2.6) becomes

4 4

R"'E,=- - =- - -

l + r^2 cosh^2 s

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