Cubic NLS on S6

The theory of the cubic NLS on the six-dimensional sphere is as follows.

Scaling is $s_c = 2\,$ .
Uniform LWP holds in $H^s\,$ for $s > 5/2\,$ BuGdTz-p3.
Uniform LWP fails in the energy class $H^1\,$ BuGdTz2002; indeed we have this failure for any NLS on $S 6$ , even ones for which the energy is subcritical. This is in contrast to the Euclidean case, where one has LWP for powers $p < 2\,$ .