This talk is based on recent joint work with G. Negro, B. Stovall and J. Tautges.

Global maximizers for the $L^2$ Fourier extension inequality on the cone in $\mathbb{R}^{1+d}$ have been characterized in the lowest-dimensional cases $d\in\{2,3\}$. We prove that these functions are critical points for the $L^p$ to $L^q$ Fourier extension inequality if and only if $p=2$. We also establish the existence of maximizers and the precompactness of $L^p$-normalized maximizing sequences modulo symmetries for all valid scale-invariant Fourier extension inequalities on the cone in $\mathbb{R}^{1+d}$. In the range for which such inequalities are conjectural, our result is conditional on the boundedness of the extension operator. The proof uses tools from the calculus of variations, bilinear restriction theory, conformal geometry and the theory of special functions.