154 20. COMPACTNESS OF THE SPACE OF K-SOLUTIONSVery recently this conjecture has been solved. Daskalopoulos, Hamilton,
and Sesum [49] have proved that any Type II ancient solution on 82 is the
King-Rosenau solution. By the work of Daskalopoulos and Sesum [50] and
one of the authors [46], a nonfiat ancient solution with bounded curvature on
JR^2 is the cigar soliton (earlier, Hamilton classified nonfiat eternal solutions^23
attaining their maximum curvature as the cigar). These works, combined
with the earlier work of Hamilton classifying nonfiat Type I ancient solutions
on surfaces with bounded curvature as having constant curvature, imply the
conjecture.5. Notes and commentary
§1. Theorem 20.1 and its proof are from §11.4 of Perelman [152]. Prov-
ing this theorem for K-solutions with Harnack rather than for K-solutions is
an unpublished observation of one of the authors [142]. Hamilton gives a
different proof of Theorem 20.1 (see subsection 2.5 of Chapter 9 of [45] for
a discussion of Hamilton's proof).
Let (wn-^1 ,g) be a Riemannian manifold and let f: (a,b)--+ IR+· The
curvature of the warped product g = dr^2 + f (r)^2 g on (a, b) x W (in par-
ticular, the Riemannian cone g 00 = dr^2 + r^2 g) may be computed as follows.
(See the solution of Exercise 1.188 on pp. 544-545 of [45].) Let {ei}r=l be a
local orthonormal frame field with en = gr and let { wj} 7=l denote the dual
coframe to { ei}· For 1 :::; i, j :::; n - 1,(20.53) (Rm 9 (ej, ei) ei, ej) = ; 2 (Rm 9 (ej, ei) ei, ej) - (~t,
(20.54)
Note that the rest of the components of Rm are zero, including(Rm 9 (ei, :r) ej, ek) = 0.
In particular, for the metric g 00 (0) = dr^2 + r^2 gw (0) defined in (20.3),
where f (r) = r so that f" (r) = O, we have for 1:::; i,j:::; n-1,
(20.55)
(Rmoo (0) (ej, ei) ei, ej) = r
1
2 \Rmgw(O) ( (ew)j, (ew )i) (ew)i, (ew )j) - : 2 ,
where ( ew )i ~ rei are orthonormal for gw, and for 1 :::; i, j :::; n - 1,
(20.56) ( Rmoo (0) ( ei,! ) :r, ej) = 0.
Note that formula (20.55) proves (19.9) whereas formula (20.56) proves
(20.4).(^23) An eternal solution is a solution which exists for all t E ( -oo, oo).