126 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONSPROOF. Since Q is a subsolution of the heat equation by (29.140), the maxi-
mum principle implies that the function t H Qmax(t) is nonincreasing.
Recall that by (29.107) we have that Voo ('if;, e) = μcos^2 'if;, where 'if; is the
latitude. Since Q vanishes on a flat cylinder, we thus have
(29.202) Q(v 00 ) =;=. V 00 I TFS(\7^3 V 00 )i2 _ =^0
on 52 - { N, S}. Geometrically, this says that Q is zero on the flat cylinder backward
limit.Suppose that the proposition is false. Then there exists c > 0 and sequences
of points qi E 52 and times ti --+ - oo such that
(29.203)By passing to a subsequence, we may assume that q 00 ~ limi-+oo Qi exists. Sincev(t) converges to v 00 in C^3 ,o. on compact subsets of 52 - {N, S} and by (29.202),
we have that q 00 is either N or S. Without loss of generality, we may assumet hat q 00 = N. Let Oi ~ O'(qi) E IR^2 , where O' : 52 - {S} --+ IR^2 is stereographic
projection. Then Oi --+ O'(N) = 0 is the origin.
Now we recall the cigar limits at the poles. Let Ki~ u-^112 (0, ti)= v^112 (0, ti)·
Since v 00 (x, y) = μ(x^2 + y^2 ), we have that Ki --+ 0. By Proposition 29.38 we havethat the sequence vi(x, y , t) ~ K i-^2 v(Kix, Kiy, t +ti), as defined in (29.116), con-
verges in C^00 on compact subsets of IR^2 to a positive function v 00 (x, y, t) satisfying
(29.118). Analogously, there is a cigar limit based at S.Case 1. The Q 2 c points are in the cigar region: lim infi-+oo l~:I < oo.
By passing to a subsequence, we may assume that Gi ~ ~ converges to a point
G 00 E IR^2 as i --+ oo. By (29.173), we have thatQi(x, y , t) ~vi la(vi)l^2 (x , y , t) = Q(Kix, Kiy, ti+ t).
In particular, Qi(Oi, 0) = Q(Oi, ti). On the other hand, we have
Qi(Gi, 0)--+ Qoo(Goo, 0) ~ Voo ia(voo)l^2 (Goo, 0) = 0,
where the last equality is true because v·~}(t)9euc is a cigar soliton. Since Q(Oi, ti)=
Q(qi, ti) by Lemma 29.50 below, we obtain a contradiction to (29.203). We conclude
that Case 1 is impossible.Case 2. The Q 2 c points are outside the cigar region: limi-+oo l~;I = oo.
The main idea is to show that rescalings of the solution about the point; (qi, ti)
subconverge to a flat cylinder, which has Q = 0.
STEP 1. A general estimate for circular averages (used to study backward