34 2. SPECIAL AND LIMIT SOLUTIONS
Moreover, the curvature R±oo ( t) at the poles is actually the maximum cur-
vature of (S^2 , g (t)) for all t < 0, since for all x > 0 we have
a a ( ,\ (coshA.t · coshx + 1) )
ax R = ax sinh (-A.t) ( cosh x + cosh A.t)
= sinhx · sinh (-A.t~ > O.
( cosh x + cosh A.t)If we compare the scalar curvature at an arbitrary point with its maximum,
we find that
. R ( t)
1
. ( cosh A.t · cosh x + 1)
hm = im = 1
t/O R±oo (t) t/O cosh (A.t) ( cosh x + cosh A.t) ·
This confirms a result equivalent to what we shall prove in Chapter 5: any
solution of the unnormalized Ricci flow on a topological 52 shrinks to a
round point in finite time.
EXERCISE 2.13. The Rosenau solution provides an example of some rel-
evance to our study of singularity models in Chapter 8: it is a Type II
ancient solution which gives rise to an eternal solution if we take a limit
looking infinitely far back in time (as described in Section 6 of that chap-
ter). In particular, there is a theorem which says that if one takes a limit of
the Rosenau solution at either pole x = ±oo as t -t -oo, one gets a copy of
the cigar soliton studied above. Demonstrate this explicitly.
REMARK 2.14. The Rosenau solution is of interest here in part because
it could potentially occur as a dimension-reduction limit of a 3-manifold sin-
gularity. (Techniques of forming limits at singularities by parabolic dilation
are presented in Section 3 of Chapter 8. The method of dimension reduction
is introduced in Section 4 of Chapter 9 and will be discussed further in thesuccessor to this volume.) Recent work of Perelman [105] eliminates this
possibility for finite-time singularities. Indeed, if the Rosenau solution oc-
curred as a dimension-reduction limit, one could by Exercise 2.13 adjust the
sequence of points and times about which one dilated in order to get a cigar
limit. Perelman's No Local Collapsing Theorem excludes this possibility.- Immortal solutions
An immortal solution of the Ricci flow is one which exists on a max-imal time interval o < t < oo, where o > -oo. The example below appears
in Appendix A of [55]. It yields a self-similar solution on JR^2 which expands
from a cone with cone angle 2?T(, where 0 < ( < 1. For t > 0, it forms
a smooth complete metric on JR^2 with positive curvature that decays expo-
nentially with respect to the distance from the origin. The reader is invited
to compare this solution with the expanding Kahler- Ricci solitons on en
described in [24] that converge to a cone as t ~ 0. (See also [40].)