16 1. RICCI SOLITONS
(1) a constant negative curvature metric on a surface transforming to
a steady soliton,
(2) an incomplete steady soliton on a surface transforming to an in-
complete metric with constant positive curvature.3.5. Classifying 2-dimensional solitons.
PROPOSITION 1.25 (Surface solitons conformal to JR^2 ). The only com-
plete steady gradient solitons conformal to the standard metric on JR^2 are
the cigar and the flat metric.
REMARK 1.26. We have not assumed the curvature is bounded or has
a sign. Note that since a steady gradient soliton is an ancient solution, by
applying the maximum principle to the evolution equation for the scalar
curvature, one sees that a complete steady Ricci soliton on a surface with
curvature bounded from below is either flat or has positive curvature (see
[111]). For a similar result to Proposition 1.25, see Corollary 1.28 below.
PROOF. Consider the pullback of the steady gradient soliton metric un-
der the map from the cylinder 51 x JR to JR^2 defined by
x = eu cosv,where ( u, v) are coordinates on 51 x JR and 51 = JR/27rZ. The pullback
is a steady gradient soliton on the cylinder, and the vector field X that
it is flowing along is conformal. Hence, the complexification^6 of X is a
holomorphic vector field and is of the form h(w)8/8w, where w = u +iv
and h(w) is a 27r-periodic entire (analytic) function. Since Xis the gradient
of a function f, then X can have no closed orbits, and hence any zeros of h
must be simple (by virtue of appealing to a local power series expansion).Consequently, critical points of f can only be local maxima and minima,
so that indexp (V f) = 1 at any critical point p. Since x ( 51 x JR) = 0, the
Poincare-Hopf Theorem, which says that the sum of the indices of V f is
the Euler characteristic, implies that f has no critical points, hence h has no
zeros. The periodicity then implies that his a constant. Moreover, X must
be a constant times a/ au, since otherwise the corresponding vector field X
on JR^2 has orbits which spiral into the origin.
Hence we know that x = ra I ar up to multiple, and then by the ar-
gument of subsection 3.1 of this chapter, J(X) = a;ae is a Killing field,
and so by Lemma 1.18 the metric on JR^2 is rotationally symmetric (and a
warped product). Then we may appeal to the above calculation of solutions
to (1.40). D
(^6) The complexification of a vector field X is defined to be x<i,o)
~ (X + iJ (X)). For example, the complexification of r gr = x gx + y gy is z gz
~ ( x gx + y gy + i ( y gx -x gy)). See the next chapter for a more detailed discussion of
Kahler manifolds (real surfaces are 1-complex dimensional Kahler manifolds). Since we
are on a surface, the complexification of a conformal vector field is a holomorphic vector
field.