l. NUMERICAL SIMULATION OF DEGENERATE NECKPINCHES 309approximate so lution. In addition to proving that there are (rotationally symmet-
ric) solutions with degenerate neckpinches, these results show that there is a wide
variety of rates of curvature blow-up- all Type Ila- to be found in such solutions.We recall that finite-time singularities may be categorized in the following way:
(I) Type I singularity: SUPMx[O,T) (T-t) IRml < oo,(Ila) Type Ila singularity: SUPMx[O,T) (T - t) IRml = oo.
Similarly, we may categorize infinite-time singularities as follows:(III) Type III singularity: SUPMx[O,oo) t IRml < oo,
(Ilb) Type Ilb singularity: SUPMx[O,oo) t IRml = oo.
Often we think of singularities forming forward in time. On the other hand,
for ancient solutions we have the following:(I-ancient) Type I ancient solution: supMx(-oo,OJ !ti IRml < oo,
(II-ancient) Type II ancient solution: SUPMx(-oo,o] !ti IRml = oo.
To this we may also add the following categorization for singularities forming
backward in time at time 0:(IV) Type IV singularity: supMx(O,T] t IRml < oo,
(Ile) Type Ile singularity: SUPMx(O,T] t IRml = oo.
We say that a singularity is Type II if it is either Type Ila, Ilb, or Ile.
PROBLEM 36.1. Given a reasonable definition of "generic", are Type II singu-
larities nongeneric? One may ask this question separately for singularities of Type
Ila, Ilb, or Ile.1. Numerical simulation of solutions with degenerate neckpinches
1.1. The role of numerical simulations.
For nonlinear partial differential equations which model physical systems- e.g.,
Navier- Stokes and Einstein's equations- numerical simulation is a familiar and
increasingly important tool for studying the behavior of solutions of the PDE. The
motivation for such study has historically come primarily from the intrinsic physical
interest of the particular solutions being simulated- e.g., ocean flow solutions of
Navier- Stokes and black hole collision solutions of Einstein's equations.
Recent experience has shown that, in addition to its effectiveness in probing the
behavior of particular solutions of some intrinsic interest (physical or otherwise),
numerical simulation can be very useful as a tool for exploring the mathematical
behavior of large classes of solutions. It has been used to seek support or to seek
counterexamples to conjectures regarding the generic behavior of solutions- e.g.,
the cosmic censorship and the Belinskii- Khalatnikov- Lifschitz conjectures for cos-
mological solutions of Einstein's equations [26]. It can also play a major role in
uncovering new, often unexpected, phenomena in families of solutions. Here, we cite
the numerical discovery by Choptuik [66] of critical behavior in sets of gravitational
collapse solutions of t he Einstein equations.
Ricci flow does not appear to model any physical systems (apart from its tie-in
with t he renormalization group from quantum field theory). There are, however, a
number of outstanding unproven conjectures regarding the behavior of Ricci flow
solutions. As we discuss here, numerical simulations have proven to be very useful
in exploring one such conjecture, and they should be useful in many others. In