24 2. SPECIAL AND LIMIT SOLUTIONS
- Eternal solutions
An eternal solution of the Ricci fl.ow is one that exists for all time.
Such solutions should have very special properties. Intuitively, one may ar-
rive at this expectation as follows. We shall see in Chapters 5 and 6 that the
curvatures of a solution to the Ricci fl.ow evolve by reaction-diffusion equa-
tions. Equations of this type involve a competition between the diffusion
term (which seeks to disperse concentrations of curvature uniformly over
the manifold as time moves forward) and the reaction term (which tends to
create concentrations of curvature as time moves forward). On an eternal
solution, one can look arbitrarily far backwards or forwards in time without
encountering any concentration phenomena. This means that such solutions
must be very stable, with no concentrations of curvature at any finite time
in either the past or the future.
2.1. The cigar soliton. Hamilton's cigar soliton is the complete Rie-
mannian surface (ffi.^2 , gL,), where
( 2 .4) gL, = dx^0 1 + dx x2 + + dy y2^0 dy
This manifold is also known in the physics literature as Witten's black
hole. (See [125].) As we shall see below, it is a steady soliton, hence
corresponds to an eternal self-similar solution. Recalling that the Christoffel
symbols I'fj of a metric g are given by
k lke(8 8 8)
rij = 2g 8xi9jf. + axJ9if. - 8xe9ij 'it is easy to compute that the Christoffel symbols of gL, with respect to the
coordinate system (z^1 = x, z^2 = y) on ffi.^2 are
rt1 = -x/ (1 + r^2 ) rb = -y/ (1 + r^2 ) n2 = x/ (1 + r^2 )
where r ~ J x^2 + y^2. Because gL, is rotationally symmetric, it is natural to
write it in polar coordinates as
(2.5)dr^2 + r^2 de^2
gL, =
1 + r^2
The scalar curvature of gL, is
(2.6)and its area form is
4
RL, = 1 + r2,1
dμL, = -- 2 dx /\ dy.
l+r
Since r^2 / (1 + r^2 ) ---> 1 as r ---> oo, equations (2.5) and (2.6) show that the
metric is asymptotic at infinity to a cylinder ofradius 1. In fact, we shall see
below that this convergence happens exponentially fast in the distance scale