3. 2-DIMENSIONAL ANCIENT SOLUTIONS WITH FINITE WIDTH 49
On the other h and, s ince the geodesic segments 'Y1 I (O,t ., 1 ,e J and -r2 I [o,t ., 2 ,e J both pass
through most of Ne: (in p articular, they both intersect '1/Jc:(S^2 x {-c^1 + 4}) and
'1/Jc:(S^2 x {c^1 - 4} )) , we have
(28.32) min{t,. 1 ,c:, t,.,,c:} 2 E-^1 rc:.
Now (28.30) follows immediately from (28.31) and (28.32). 0REMARK 28.31. Theorem 28.30 may also b e proved using Corollary 9.88 in
Morgan and Tian [251].In contrast to Theorem 28.30, in dimension 4 it is exp ected that t he asymptotic
cones of singularity models can b e top-dimensional. For example, under the Kahler-
Ricci flow of complex surfaces, consistent with Mori's minimal model program, it
is exp ected tha t one of the shrinkers of Feldman, Ilmanen , and one of t he authors
[111] occurs as a singularity model.3. Noncompact 2-dimensional ancient solut ions with finite width
Singularity models, whether they b e compact or noncompact, which correspond
to the Ricci flow on closed manifolds a re necessarily K:-noncollapsed at all scales.
Moreover , those singularity models which dimension reduce to dimension 2 have
b een classified by the works of Hamilton and P erelman as having the round shrink-
ing 2-sphere (or its Z2 quotient) as their nonflat factor. This uses the fact that for
each K: > 0 the cigar is K:-colla pse d at all large enough scales. However , it is still of
substantial interest to understand ancient solutions which are not K:-noncollapsed
for any K: > 0.
In this section, all the res ults shall p ertain to 2-dimensional complete ancient
solutions with bounded curvature. A res ult of Ha milton (see Theorem 9.14 in [77])
states that, in the T ype I case, the universal cover of any such solution must be
either the Euclidean plane or the shrinking round 2-sphere. Hence we co nsider the
T yp e II case. By t he work of one of the authors (Sun-Chin Chu) in [83], any such
noncompact Type II ancient solution must have finite width (see Definit ion 28.39
b elow). Furthermore, Panagiota Daskalopoulos and Natasa Sesum proved that a ny
noncompact Type II ancient solution with finite width is isometric to a multiple of
the cigar; this is the result we presently discuss (see Theorem 28.41 b elow).
In the remaining case of a compact Type II ancient solution, Daskalopoulos,
Ha milton, and Sesum proved t h at it must b e isometric to a multiple of t he King-
Rosen au solution; we discuss this result in the next chapter. The above works
combine to give us a complet e classification of 2-dimensional co mplete ancient so-
lutions to the Ricc i flow with bounded curvature.
3.1. Relative isoperimetric inequalities.Let (M'', g) b e a Riema nnia n manifold and let Vol denote the Riemannian
measure for measurable subsets. Given a n ( n - 1 )-dimensional C^00 submanifold
Nn-^1 , let Area(N) denote its (n - 1)-dimensional volume with resp ec t to the
induced metric.
DEFINITION 28 .32. We say that a sequence of measurable sets {Ei} in (Mn,g)volume converges to a measurable set E if limi-+oo Vol(Ei 6 E) = 0, where
A 6 B = (A - B) U (B - A) denotes the symmetric difference of two subsets A
and B.