l. NUMERICAL SIMULATION OF DEGENERATE NECKPINCHES70
60
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10
00.08 0.1
0.04 0 .06- 02
FIGURE 36.5. Rs2 for supercritical Ricci flow.20
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-150.08 0.1
0.04 0.06
0. 02FIGURE 36.6. R.l.. for supercritical Ricci flow.0.16 0.18
0.12 0.140.16 0.18
0.12 0.14315poles. At the poles, both curvatures get very la rge. In a sense, the geometry
approaches that of a 3-dimensional javelin (i.e., a long thin cylinder with pointed
ends), with S^2 cross sections. This is the conjectured behavior for a degenerate
neckpinch.
As noted above, it has been further conjectured that the behavior of the flow
at the poles for critical initial data is locally modeled by the Bryant steady soliton
solution. More specifically, the idea is that if tk is a sequence of times approaching
the singular time T and if each snapshot of the flow g(tk) is dilated in a neighbor-
hood of the pole in such a way that the curvature is normalized to unity at the
pole, then the dilated metrics g(tk) locally (about the pole) approach the geometry
of the Bryant steady soliton.
In the numerical work discussed in [117], this conjecture is tested by examining
the Ricci flow of a sequence of subcritical initial metrics (i.e., a sequence of Ai
parametrized geometries with Ai approaching 0.1639 from above). For each of
these flows one seeks the time ti at which the maximal curvature is achieved on the
pole, one dilation-normalizes each 9>., (ti) as described above, and then one compares