10. AN UNRESCALED CIGAR BACKWARD CHEEGER-GROMOV LIMIT 107
at the point N and using the diffeomorphisms Wi ~ cr-^1 o <I>i o er, converge in the
pointed C^00 Cheeger- Gromov sense to a cigar soliton g 00 (t) ~ v;:;;,}(t)geuc·
The proof of this proposition occupies the rest of the section. By (29.24), thereexists a constant C < oo such that
(29.120) 0 < Rgi(t):::; C fort E (-oo,-1-ti].
We have the following rough, but uniform, estimate.LEMMA 29.39. There exists C < oo such that on ffi.^2 ,
(29.121) -C::::: lnvi(r,e,o)::::: c (1 +r^2 )
in polar coordinates.
PROOF. Similarly to as in the proof of Lemma 29.32, we definewi(r) ~ ( lnvi(r,e,O)de.
}51By (29.117), we have
(29.122) O < ( .6.euc ln vi (r, e, 0) de
}51
= ( ~~ (rBlnvi) (r,e,O)de
}51 r 8r 8r= ~!£ (rdwi(r)).
r dr drNow by (29.34) we have that limr--+O r~(r) = 0. Therefore r~(r) > 0 for
r > 0, which implies that r H wi(r) is a strictly increasing function. By this and
vi(O, 0) = 1, we obtain
(29.123) ( lnvi(r,e,O)de > 0.
}51
Rewriting (29.113) using ii= i(l + r^2 )^2 v o cr-^1 and (29.116), we obtain
(29.124) (^8 - )
2
ae ln iii :::; C on ffi.^2 x ( -oo, -1 - ti].This and (29.123) imply that lnvi(r,e,o) 2: -C.
By (29.122), (29.117), and (29.120), we then have
~ dd (r ddwi (r)) = r 1:g; (r, e, 0) de::::: c
r r r } 51 vifor some constant C < oo. Since limr--+or~(r) = 0, integrating this yields
~(r):::; ~r. Since wi(O) = 27rlnvi(O,O) = 0, we obtain that wi(r):::; !j.r^2. We
conclude by (29.124) the upper bound in (29.121), which proves the lemma. D
Now, by (29.117), (29.121), and (29.120), we obtain that there exists a constant
C < oo such that for any T E ( 1 , oo) and any i sufficiently large so that ti + T :::; -1,
(29.125) -CT:::; ln vi (r, e, t) :::; C(T + r^2 ) on ffi.^2 x [-T, T].
By standard parabolic estimates similar to those in the proof of Lemma 28.53,
we obtain uniform estimates for the higher derivatives of Vi on compact subsets of
ffi.^2 x ( - oo, oo). Thus, by the Arzela- Ascoli theorem, there exists a subsequence