  We show that, possibly after a compactification of spacetime, constant
functions are local maximizers of the Tomas-Stein adjoint Fourier restriction
inequality for the cone and paraboloid in every dimension, and for the sphere
in dimension up to 60. For the cone and paraboloid we work from the PDE
framework, which enables the use of the Penrose and the Lens transformations,
which map the conjectured optimal functions into constants.
