Existence of extremals for a Fourier restriction inequality on the one-sheeted hyperboloid

We prove the existence of functions that extremize the endpoint $L^2$ to $L^4$ adjoint Fourier restriction inequality on the one-sheeted hyperboloid in Euclidean space ${\mathbb{R}}^4$ and that, taking symmetries into consideration, any extremizing sequence has a subsequence that converges to an extremizer.