26 2. SPECIAL AND LIMIT SOLUTIONS
In particular, R"B = 0 ( e-^2 s) as s ---7 oo, which illustrates the exponential
decay claimed above.
We will now show that the cigar is essentially unique. The argument
will also construct the potential function, hence the gradient vector field
that makes g"B into a soliton.
LEMMA 2.7. Up to homothety, the cigar is the unique rotationally sym-
metric gradient Ricci soliton of positive curvature on JR^2.PROOF. Without loss of generality, one may write any rotationally sym-
metric metric g on JR^2 in the form
(2.8) g = ds^2 + <.p (s)^2 de^2 ,
where <.p (s) is a strictly positive function. (Compare with equation (2.7)
above.) We shall calculate using moving frames, as will be reviewed in
Section 1 of Chapter 5. It is natural to write
g = 15ij wi @ wj
in terms of the orthonormal coframe field { w^1 , w^2 } given by
w^1 = ds, w^2 =1.p(s) de.
The orthonormal frame field { e 1 , e2} dual to { w1, w^2 } is then given by
a 1 a
ei = as' e^2 = <.p ( s). ae.It is straightforward to check that for any vector field
X=X^1 ~ X^2 ~as+ ae
on JR^2 , one has
i.p^1 ( S) 2 a /V' xe1 = <.p (s) X ae = <.p (s) de (X) · e2
V' xe2 = -<.p' ( s) X^2 :s = -<.p' ( s) de (X) · e1.
It follows that the Levi-Ci vita connection 1-forms { w{} defined with respect
to{e1,e2}by.
V' xei = w{ (X) · ej
have only two nonzero components:
(2.9) wi = -w~ = <.p^1 (s) de.
In general, Cartan's first and second structural equations are
d w i =w J. /\wj i an d Hi nJ -_ d wi j - wi k /\wk. jIn the case at hand ' we have dw^1 = (^0) ' dw^2 = 'P'(s) <p(s) w^1 /\ w^2 ' and
ni = dwi = '{J^11 (s) ds /\de= <.p" (s) w^1 /\ w^2.
<.p (s)