- CHARACTERIZING ROUND SOLUTIONS 91
to (N 00 ,9 00 (0),p 00 ) that Oi converges to a point 000 E loo C N 00 and that /3Hsi)
converges to a tangent vector V 00 E T 000 N 00 which is unit with respect to 900 (0).
This implies that ai ( s ) ~ j3 (Si + s) converges to a path a 00 : ( -oo, oo) ---t N 00 , with
a:X,(O) = V 00 , which is a geodesic line with respect to 900 (0).
Since 900 (0) has nonnegative curvature, by the Aleksandrov splitting theoremwe obtain that (N 00 , 900 (0)) splits off a line. Since we are in dimension 2 and since
N 00 contains the embedded geodesic loop 100 with respect to 900 (0), we conclude
that (N 00 ,9 00 (0)) is a flat cylinder JR x 51 (r 00 ) with a product metric. Hence,
by translating the s coordinate, we may assume that loo = {O} x 51 (r 00 ). This
completes the proof of Lemma 29.24.
8.3. Bounding the areas of B^9 x, ~)(Pt) by time.
In this subsection we further explore how any minimizer It of I ( ·; 9(t)) divides
52. For this purpose, we first prove results about c-necks. To garner some intuition,
we first consider the flat cylinder (JR x 51 , 9cyi). In this case, for any points ( s 1 , 81 )
and (s2, 82), we have
Hence
(29.75) sup d 9 cyI ((s1, 81), (s2, 8)) - inf dgcyI ((s1, 81), (s2, 8))
llES' llES^1
= V(s2 - s1)2
+ 71'^2 - ls2 - s1I
71'2<.
-21s2-s1I
From this we see that if a~ ls2 - s1I ~ N, then
(29. 76) { s 2 } x 51 C B(::^1 ,o;) (a+ ;~) - B(;::o,) (O') for each 81, 8~ E 51.
Similarly, for s 2 = s 1 + O' and where O' ~ N, if we define 1 to be the component of
oB(;::o,)(O') close to {s2} x 51, then
(29.77) 1 C [ s2 - ~ , s2] x 51 provided N ~ 71'.Recall that a normalized €-neck NE: in a Riemannian 2-sphere ( 52 , 9), cen-tered at a point p E 52 , is given by an embedding cp : BP ( c-^1 ) ---t JR x 51 with
cp(p) E {O} x 51 and such that
lcp*(ds^2 + d8^2 ) - 9icr,-1 1 +i(g) < c.
Let (s,8) = (s(cp(x)),8(cp(x))) be the coordinates on Bp (c-^1 ) induced by the
c-neck structure. We shall also denote Is ~ cp-^1 ({s} x 51 ) for s E (-c-^1 + 4,
c^1 - 4). Each embedded loop Is divides 52 into two open disks 5'},+ and 5'},_,
where cp-^1 ((s, ±(c^1 - 4)) x 51 ) c 5'1,±·