The theory of the cubic NLS on the six-dimensional sphere is as follows.
- Scaling is .
- Uniform LWP holds in for BuGdTz-p3.
- Uniform LWP fails in the energy class BuGdTz2002; indeed we have this failure for any NLS on , even ones for which the energy is subcritical. This is in contrast to the Euclidean case, where one has LWP for powers .