314 36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES
18
16
14
12
10
8
4
2FIGURE 36.3. Rs2 for subcritical Ricci fl.ow.18(^16 14)
12
10
8
6
4
2
0
-2
ljl 0.2
0.4
1
0.6^0 .8
FIGURE 36 .4. Rl. for subcritical Ricci fl.ow.
evolution of the Ricci curvature eigenvalues^3 Rs2 and Rl. which are found to graph
out as shown in the Figures 36.3 and 36.4.
For the "small value" 0.1 of .A corresponding to tight pinching (supercritical
fl.ow), we find that in the neighborhood of the equator, Rs2 grows without bound
as t increases, while Rl. stays bounded. As well, away from the equator, both are
bounded; see Figures 36.5 and 36.6.
This signals the formation of an S^2 n eckpinch singularity at the equator.
The "critical value" for A is found by numerical experiment. That is, one
examines the behavior of the fl.ow for values of .A between 0.1 and 0 .2 and finds
neckpinching singularity behavior for all .A < 1.639, while the geometry flows to a
round sphere for A > 1.639. For the fl.ow starting with A = 0.1639, the behavior
is markedly different. There, as seen in Figures 36.7 and 36.8, Rl. gets small
everywhere except at the poles, while Rs2 slowly grows everywhere except at the
(^3) These are calculated to be
(36.9) Rl. = -2e^2 (W-X) (-1 + X" + W" + (X' + 3W')cot1j> + 2(X' + W^1 )W^1 ],
[
1 e-4W
R 82 = -e^2 (W-X) -2 + -. 2 + X" + W" + (3X' + 5W') cot1j>
sm 1f>
(36.10) + (X' + W')(X' + 3W')](where we note that W, W' and W" can be expressed in terms of S, S' and S").