Cubic NLS on S6

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

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