1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

1. NUMERICAL SIMULATION OF DEGENERATE NECKPINCHES 313

At the ghost zones one uses smoothness of the metric, which implies that X' and

S' vanish at the poles. This condition may be implemented as X 1 = X 2 and

XN = XN-1 (and correspondingly for S).

1.2.3. Numerical results.
To study neckpinching and the related critical behavior, the numerical evolution
outlined a bove was carried out and reported in [116] for a wide range of the >--
parametrized neckpinch initial data, as is made explicit in Subsection 2.2 of that
work. The behavior observed in these numerical simulations matches that which
had been speculated. In particular, the following is seen.
For the "large value" 0.2 of the pinching parameter >- (subcritical flow), one
finds that the geometry evolves towards the round sphere geometry. This is seen in
the graphs in Figure 36 .1 and Figure 36.2, which show X approaching a constant
and S approaching 0. One also sees this asymptotic behavior by tracking the

0.3

0 .2

0 .1

0

-0.1

-0.2

-0.3

·0.4

-0.5

·0.6

FIGURE 36. l. X for subcritical Ricci flow.

·0.2

·0.4

-0.6

·0.8

-1

-1.2

-1.4

-1.6

FIGURE 36.2. S for subcritical Ricci flow.