- CONSTRUCTING THE BRYANT STEADY SOLITON 17
We have the following from Theorem 13 on p. 61 and Theorem 10 on p.
55 of Huber [210] (see also §10 of Li [251]).THEOREM 1.27 (Conformal structure of surface with finite total curva-
ture). If (M^2 ,g) is a complete Riemannian surface with JM K_dμ < oo,
where K_ ~max {-K, O}, then (M, g) is conformal to a closed Riemann-
ian surface with a finite number of points removed. Furthermore, by the
Cohn-Vossen inequalityJM K+dμ ::; JM K_dμ + 2nx (M) < oo,
where K+ ~ max{K,O}.By the first part of the above theorem, a complete noncompact surface
with positive curvature, which we know is diffeomorphic to the plane, must
be conformal to the plane. From this and Proposition 1.25 we conclude the
following.
COROLLARY 1.28 (Steady surface soliton with R > 0 is cigar). If (M^2 , g)
is a complete steady gradient Ricci soliton with positive curvature, then
(M,g) is the cigar.This result gives us a classification of complete steady Ricci solitons on
surfaces with curvature bounded from below since such solutions are either
flat or have positive curvature.
4. Constructing the Bryant steady soliton
We may generalize the cigar metric to a rotationally symmetric steady
gradient Ricci soliton in higher dimensions on JRn+l by setting N = sn,
the unit sphere with constant sectional curvature + 1.^7 As the following
calculations parallel unpublished work of Robert Bryant for n = 2, we will
refer to the complete metrics obtained as Bryant solitons. The Bryant
soliton is a singularity model for the degenerate neckpinch, a finite time
singularity which is expected to form for some (nongeneric) initial data on
closed manifolds. (See Section 6 in Chapter 2 of Volume One.)
REMARK 1.29. A singularity model is a long existing solution (i.e.,
the time interval of existence has infinite duration) of the Ricci flow obtained
from a singular solution (i.e., a solution defined on a maximal time interval
[O, T)) of the Ricci flow as the limit of dilations about a sequence of points
and times approaching the singularity time T.
7For background on Einstein metrics which are warped products over 1-dimensional
bases, see the notes and commentary at the end of this chapter.