1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. WARPED PRODUCTS AND 2-DIMENSIONAL SOLITONS 11

Sections 4.3 and 5 of Chapter 3 of [108]). Once this condition is added to

(1.17), the system becomesinvolutive (see [221] for the proof). Furthermore,

involutivity implies that Cauchy problems for the augmented system (1.17)

and (1.36) are locally solvable; for example, the 1-jet of g may be arbitrarily

prescribed along a hypersurface transverse to a given vector field X. Thus,

Ricci soliton structures (g, X) locally depend, modulo diffeomorphisms, on

n(n + 1) functions of n - 1 variables (i.e., the components of g, and their

transverse derivatives, along the hypersurface).

3. Warped products and 2-dimensional solitons

In this section, we review some of the examples of 2-dimensional Ricci
solitons, such as the cigar, which have been constructed to date; the Bryant
soliton is discussed in Section 4 of this chapter and examples of Kahler-Ricci
solitons will be discussed in the next chapter. In Section 1 of Appendix B
we will give some conditions under which the Ricci fl.ow converges to some
of the solitons discussed here.
Recall from Lemma 5.96 on p. 168 of Volume One that any ancient
solution of the Ricci fl.ow on a surface of positive curvature that attains its
maximum curvature in space and time is the cigar. Similarly, any ancient
solution of the Ricci fl.ow on a manifold of positive curvature operator that
attains its maximum curvature in space and time is a steady gradient Ricci
soliton. In dimension 3, it is conjectured that such a soliton is the Bryant
soliton. This is one reason for focusing our attention on the cigar and Bryant
solitons.

3.1. Solitons and Killing vector fields on surfaces. If (g, X) is a

soliton structure on a surface M^2 , then X is a conformal vector field. This

is simply because \7 iXj + \7 j Xi = -( R + c:) gij. If (g, \7 f) is a gradient

soliton, then J (\7 f) is a Killing vector field (where J : TM --t TM is the

complex structure, defined as counterclockwise rotation of tangent vectors

by 90°). To see this, we observe that since \7 f is a conformal vector field on

a surface,

(.CJ(\Jf)g)ij = \7i (1j\7kf) + \7j (1f\7kf) = Jj\7i\7kf + Jf\7j\7kf

= - ( ~ + ~) (Jjgik + Jfgjk) = o,

where the components Jj are defined by J ( 8 ~i) ~ Jj 8 ~j. That is, J (\7 f)

is a Killing vector field.

In dimension 2 we have the following.

LEMMA 1.18. A surface with a Killing vector field is locally a warped

product. In particular, a gradient soliton on a surface is locally a warped

product.