316 36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES
160
140
120
100
80
60
40
20
FIGURE 36. 7. R 52 for critical Ricci flow.
160
140
120
100
80
60
40
20
0
-20
FIGURE 36.8. Rl. for critical Ricci flow.
1.2
1.2
with the geometry of the Bryant steady soliton. As shown in Figures 36.9 and 36.10
(note that the so lid line is the critical solution at the time of maximum curvature
and the dashed line is the Bryant steady soliton), for Ai very close to the critical
value, the match is striking.
1.3. Future numerical work and conclusions.
The numerical work described below in Subsection 2.2 of this chapter is far
from complete. It is important that a much wider variety of parametrized families
of initial data-all having the feature that for one extreme of the parameter there is
clearly a neckpinch singularity forming during the flow while for the other extreme
the flow approaches the round sphere geometry- be used as starting points for
the Ricci flow evolution. Included should be families which are not rotationally
symmetric. Is there always a unique transition value for the parameter, dividing
pinching from converging flows? Does one always find the same sort of very special
behavior- formation of a javelin-type geometry, with the curvature concentrating