310 36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES
addition, with so much of the b ehavior of Ricci flow not yet understood, such
exploration is likely to uncover new and surprising phenomena in Ricci flow.
While there h as b een much discussion of possible numerical simulation of Ricci
fl.ow solutions (it has b een listed as a topic for discussion at many Ricci flow work-
shops during the p ast several years), to date, very few such simulations have b een
carried out in a serious way. One of the very few questions which h as b een explored
vi a numerical si mulation is that of the Ricci fl.ow of critical geometries (in the sense
defined above) and whether these flows produce Type II singularities and the conjec-
tured characteristic degenerate neckpinch behavior. This work [116], [117], which
was largely carried out b efore the Gu and Zhu proof appea red, strongly indicates
that (rotationally symmetric) critical geometries do produce T yp e II singularities,
with degenerate neckpinch behavior.
Apart from the role of this numerical study in extending our underst a nding of
the geometries whose Ricci flows develop T yp e II singularities and in providing the
details of the formation of these singularities, this work provides a useful example
of what numerical simulation can and cannot do in exploring mathem atical conjec-
tures. We note especially that these numerical simulations have proven to be very
useful in guiding subsequent an alytical work detailing the formation of degenerate
neckpinch singularities, as described b elow.
One feature of many of the conjectures concerning Ricci fl.ow which has to some
extent hindered their exploration by numerical simulations is that in most cases the
evolving geometries under study live on manifolds with nontrivial topology. This
is an issue beca us e the vast majority of numerical techniques have been developed
for use on manifolds which are subsets of !Rn. We note, however , recent work [205]
which h as developed numerical techniques for use on ma nifolds with nontrivial
topology.
1.2. Setting up and carrying out the numerical simulation.
We now focus on the work done in [116] and [117], which uses numerical
simulation to study Type II rotationally symmetric degenerat e neckpinches.
1.2. l. Initial data and evolution equations for rotationally symmetric critical
neckpinch solutions.
Numerical explorations of the behavior of solutions of PD Es involve the explicit
numerical construction of (approximations to) particular solut ions to the PDE sys-
t em. Hence the first step in such a n exploration is to identify and parametrize
the solutions to b e considered and to write out the PDE system in t erms of the
functions appearing in those solutions. We describe now how this is done for the
critical neckpinch study.
While it would be very useful to explore what happens for geometries which are
not rota tionally symmetric, the first stage of the critical neckpinch studies focuses
on geometries with this strong assumpt ion of isometry. Such metrics on S^3 may b e
written in the form
(36.l)where ('I/;,(}, </>) gives the standard angular coordinates on the three sphere and
where spherical symmetry holds as long as the functions X and W are functions of
'I/; only. To also ensure regularity of the geometry at the poles (which are m arked by
the coordinate values 'I/; = 0 and 'I/; = 7r), one requires that the quantity S ~ si~ ..μ