- DECOMPOSITION THEOREMS AND THE RICCI FLOW 441
regions with curvature comparable to their spatial maximum are geomet-
rically close to a 3-dimensional steady Ricci soliton, whereas some regions
with curvature not comparable to their spatial maximum are geometrically
close to a shrinking round product cylinder.
In [189] Hamilton developed surgery theory and formulated a version of
Ricci fl.ow with surgery. Although this theory was developed for solutions
on closed 4-manifolds with positive isotropic curvature, its higher aim was
clearly a surgery theory for 3-manifolds. Indeed, the class of 4-manifolds
with positive isotropic curvature is flexible enough to allow for connected
sums. Many of the techniques developed in [189] applied to the setting of
the Ricci fl.ow on closed 3-manifolds. Limiting arguments and the study
of ancient solutions were developed by Hamilton with the aim of enabling
surgery. A contradiction argument using limiting techniques was proposed
to show that for suitable surgery parameters, the set of surgery times is
discrete, and in particular, do not accumulate in finite time. Unfortunately,
as was known to some mathematicians working in the field of Ricci fl.ow
and as pointed out in [298], there was an error in this part of Hamilton's
argument.
In the recent work of Perelman [297], [298], building on Hamilton's the-
ory, Ricci flow behavior (especially singularity formation) on 3-manifolds is
carefully examined and classified. The overall picture is subtle and tech-
nical, with some of the foundations being discussed in this volume. In the
following two simplified examples (the first of which continues our discussion
above), we try to convey some of its topological flavor, omitting most of the
details.
The formation of neckpinch singularities (as described above) in a certain
sense reflects the topological connected-sum decomposition of the underlying
manifold. Indeed, suppose a neckpinch with two ends occurs on a region
identified with 52 x R Hamilton proposes a surgery process as follows
[189].^4 One does surgery near the large ends of the long, thin tubes in
that part of the manifold identified with 52 x IR, capping these off with
round 3-balls. Note that Hamilton's theory predicts the existence of such
tubes where the curvature is very large at the center and slowly decreases as
one moves away from the center along the relatively very long length of the
tube. In fact his theory predicts that as one approaches the singularity time,
the tube becomes arbitrarily close to an exact cylinder and its size slowly
increases as one moves away from the center to an arbitrarily much larger
but still very small size.^5 Note that Perelman's surgery process in [298] is
a modification of the surgery process proposed earlier by Hamilton.^6
(^4) More precisely, he considers the 4-dimensional version of this.
5More precisely, the tube is conformally close to a round product cylinder, where the
conformal factor changes very slowly as one moves away from the center.
(^6) Huisken and Sinestrari have considered an analogue of Hamilton's surgery theory for
the mean curvature flow.