On the extremizers of an adjoint Fourier restriction inequality

The adjoint Fourier restriction inequality for the sphere $S^2$ states that if $f\in L^2(S^2,\sigma )$ then $\stackrel{?}{f\sigma }\in L^4\left({\mathbb{R}}^3\right)$. We prove that all critical points $f$ of the functional ${6\stackrel{?}{f\sigma }6}_{L^4}/{6f6}_{L^2}$ are smooth, any complex-valued extremizer for the inequality is a nonnegative extremizer multiplied by the character $e^{ix?\xi }$ for some $\xi$, and complex-valued extremizing sequences for the inequality are precompact modulo multiplication by characters.