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104 19. GEOMETRIC PROPERTIES OF K;-SOLUTIONS

Note that the King-Rosenau solution is qualitatively different from Perel-

man's ,,;-solution. This is related to the fact that the cigar soliton and the

Bryant soliton are qualitatively different (for example, the curvature of the

cigar soliton decays exponentially whereas the curvature of the Bryant soli-

ton decays inverse linearly).

For further discussions related to Perelman's ancient solution on 5n, see

Example 20.12 and Optimistic Conjecture 20.26.

Finally we remark that there is also a construction of Perelman's rota-

tionally symmetric noncompact (not ancient) standard solution on :!Rn (see

§2 of [153]).

4. Equivalence of 2-and 3-dimensional ,,;-solutions with and

without Harnack

In this section we shall show that, in dimensions 2 and 3, the notion of

,,;-solution with Harnack is equivalent to the original notion of ,,;-solution;

this result is due to one of the authors [142].

4.1. Classification of 2-dimensional ,,;-solutions.

The following classification result, due to Hamilton (see §26 of [92]), is

also stated as Corollary 11.3 in [152] (see also Corollary 9.19 in [45] for a

proof).

THEOREM 19.42 (Any 2-dimensional ,,;-solution is 5^2 ). There is only one

orientable 2-dimensional ,,;-solution: the shrinking round 2-sphere.

We may prove the following generalization.

COROLLARY 19.43 (Any 2-dimensional ,,;-solution with Harnack is 52 ).

There is only one orientable 2-dimensional ,,;-solution with Harnack: the

round shrinking 52.

PROOF. By Theorem 19.42, it suffices to prove that any 2-dimensional
,,;-solution with Harnack has bounded curvature in space-time; we prove
this by contradiction. Suppose that there exists a ,,;-solution with Harnack

(M^2 ,g(t)), t E (-oo,O], which does not have bounded curvature in M x

(-oo, 0]. Then
sup R (x, 0) = oo
xEM
(using the fact that the trace Harnack estimate implies ~f 2: 0).
Fix a point p EM. Since supxEM R (x, 0) = oo, we may apply Proposi-
tion 18.12, so that there exists a sequence of points {xi}~ 1 and a sequence
of positive numbers {ri}~ 1 such that

(19.39) d 9 (o) (xi,P)--+ oo, R(xi,O)rl--+ oo, dg(O) (xi,P) /ri--+ oo,

and

(19.40) sup R (x, 0) ::; 4R (xi, 0).
Bg(O) (xi,ri)