- THE EQUIVALENCE OF Q AND Q 129
The annulus ([-1, 1] x 51 , gi(O)) is isometric to (Bo(es,+^1 ) - B 0 (e^8 '-^1 ), g(ti )). So
we h ave
( Rg(t,)v-^1 (ti)dμeuc;:::: J.
} Bo(e^5 i+^1 )-Bo(e•i-^1 )
On t he other hand , since gi(t) = vi^1 (t)9euc co nverges to a cigar soliton (this is
the cigar limit b ased at N), for a ny 'T/ > 0 there exists r'l < oo such that
( R9i(o)vi^1 (O)dμeuc > 47r - 'T/·
} Bo(r.,)
The Riemannian ball (Bo(r'1), gi(O)) is isometric to (Bo(Kir1J), g(ti)), where g(t) ~
v(t)9euc· So we h ave
( R 9 (ti)v-^1 (ti)dμeuc > 47r - 'T/·
} Bo(Kir.,)
Simila rly, the cigar region near S yields integral scalar curvature greater than 47r-'T).
More precisely, there exist f'i ---+ oo such that
( R 9 ct,)v-^1 (t i )dμeuc > 47r - 'T/·
j!R^2 - Bo(i\ )
Recall that Ki ---+ 0, e^8 i ---+ 0, and ~;; ---+ oo. Therefore the sets Bo(K ir, 7 ) , IR^2 -
Bo(ri ), Bo(e^8 '+1) - Bo(e^8 '-^1 ) a re disj oint and we conclude that
87r = ( R 9 (ti)dμ 9 (ti) > 87r - 2'T) + J > 8 7r
J s 2
by choosing 'T/ < J / 2, which is a contradiction. This completes t he proof of Claim 2.
By Claim 2 and the strong maximum principle, we must have R9 00 = 0 on
!Rx 51 x(-oo, oo). From (29.7), wethenobtainb.cy1lni\x, = 0. Sincevi (s, B, t) > μ ,
we h ave the global l ower bound ln v 00 ( s, B) ;:::: ln μ. Lifting ln v 00 from IR x 51 to IR^2 ,
we may apply the Liouville theorem for entire harmonic functions that are bounded
from below to conclude that ln v 00 is constant. Thus v-;:;,^1 ( t )9cyl is a flat cylinder.
STEP 4. Case 2 is impossible. Now let
Vi(r, B, t) ~ r^2 vi (ln r , B, t) = v(ln r +Si, B + B;, t +ti).
Since v;;;,1 (0)9euc: where v 00 ~ limi-;oo vi, is a flat cylinder, we h ave
fan Q(vi)(l, 0, 0) = Q(voo)(l, 0, 0) = 0.
i-;oo
On the other hand , Q(vi)(l, 0, 0) = Q(v)(qi, ti ) 2: c:; we obtain a contradiction.
Si nce Case 1 and Case 2 are impossible, the proposition is proved. 0
16. The equivalence of Q and Q
Let <J : 52 - {S} ---+ IR^2 denote stereographic projection and let <p(x, y) ....:...
(x2+:2+l)2. Let g = v-^1 gs2 ~ <J*(v-^1 9euc) b e a Riemannia n m etric on 52. By
(29.6), on JR^2 we h ave v o <J-^1 = <pv. Let Q and Q be as defined in (29.134) and
(29.1 7 2), resp ectively.
We h ave the following.
LEMMA 29.50 (Equivalence of Q and Q). We have Q o <J-^1 = Q.
