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 $H^{1}\,$ BuGdTz-p2; 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 $p<2\,$.